Journal of Logic and Computation - recent issues

Connections between Belief Revision, Belief Merging and Social Choice

Gabbay, D., Rodrigues, O., Pigozzi, G. — 2009-05-21

Geodesic Revision

Georgatos, K. — 2009-05-21

The purpose of this article is to introduce a class of distance-based iterated revision operators generated by minimizing the geodesic distance on a graph. Such operators correspond bijectively to metrics and have a simple finite presentation. As distance is generated by distinguishability, our framework is appropriate for modelling contexts where distance is generated by threshold, and therefore, when measurement is erroneous.

Aggregating Judgements by Merging Evidence

Williamson, J. — 2009-05-21

The theory of belief revision and merging has recently been applied to judgement aggregation. In this article I argue that judgements are best aggregated by merging the evidence on which they are based, rather than by directly merging the judgements themselves. This leads to a three-step strategy for judgement aggregation. First, merge the evidence bases of the various agents using some method of belief merging. Second, determine which degrees of belief one should adopt on the basis of this merged evidence base, by applying objective Bayesian theory. Third, determine which judgements are appropriate given these degrees of belief by applying a decision-theoretic account of rational judgement formation.

Aggregating Partially Ordered Preferences

Pini, M. S., Rossi, F., Venable, K. B., Walsh, T. — 2009-05-21

Preferences are not always expressible via complete inear orders: sometimes it is more natural to allow for the presence of incomparable outcomes. This may hold both in the agents' preference ordering and in the social order. In this article, we consider this scenario and study what properties it may have. In particular, we show that, despite the added expressivity and ability to resolve conflicts provided by incomparability, classical impossibility results (such as Arrow's theorem, Muller–Satterthwaite's theorem and Gibbard–Satterthwaite's theorem) still hold. We also prove some possibility results, generalizing Sen's theorem for majority voting. To prove these results, we define new notions of unanimity, monotonicity, dictator, triple-wise value-restriction and strategy-proofness, which are suitable and natural generalizations of the classical ones for complete orders.

Non-manipulable Social Welfare Functions when Preferences are Fuzzy

Perote-Pena, J., Piggins, A. — 2009-05-21

It is well known that many social decision procedures are manipulable through strategic behaviour. Typically, the decision procedures considered in the literature have been social choice correspondences. In this article, we investigate the problem of constructing a social welfare function that is non-manipulable. In this context, individuals attempt to manipulate a social ordering as opposed to a social choice.

Using techniques from fuzzy set theory, we introduce a class of fuzzy binary relations of which exact binary relations are a special case. Operating within this family enables us to prove an impossibility theorem. This theorem states that all non-manipulable social welfare functions are dictatorial, provided that they are not constant. A proof of this theorem first appeared in Perote-Peña and Piggins (2007, J. Math. Econ., 43, 564–580). This article contains a new proof of this theorem which is considerably simpler than the original. Moreover, we also consider a possibility result which this earlier article neglects.

A General Approach to Aggregation Problems

Daniels, T. R., Pacuit, E. — 2009-05-21

We discuss a general approach to judgement aggregation based on lattice theory. Agents choose elements of a lattice, and an aggregation procedure yields a ‘social choice’ based on the individual choices. Settings traditionally studied in social choice theory can be thought of as implicational systems, and lattice theory provides an abstraction of such systems. In fact, traditionally studied settings correspond to certain atomistic lattices in our framework. Our aim is to systematically investigate how properties of a given lattice induce constraints on aggregation procedures that lead up to impossibility theorems. We allow for non-atomistic lattices and this raises some subtle issues. We will discuss how well our framework fits in with the traditional approaches to social choice theory, in particular with respect to generalizations of some of the well known axioms, and go on prove an impossibility result that highlights the role of certain lattice theoretical properties. These properties reflect some of the traditional axioms or other aspects of traditional systems.

Complexity Issues in Axiomatic Extensions of Lukasiewicz Logic

Cintula, P., Hajek, P. — 2009-03-26

In this article, the computational complexity of all axiomatic extensions of Lukasiewicz propositional logic L and the arithmetical complexity of both the general and standard semantics of their corresponding predicate logics is determined.

Rewriting Corner

Fernandez, M. — 2009-03-26

Paramodulation with Well-founded Orderings

Bofill, M., Rubio, A. — 2009-03-26

For many years, all existing completeness results for Knuth–Bendix completion and ordered paramodulation required the term ordering to be well-founded, monotonic and total(izable) on ground terms. Then, it was shown that well-foundedness and the subterm property were enough for ensuring completeness of ordered paramodulation. Here we show that the subterm property is not necessary either. By using a new restricted form of rewriting, we obtain a completeness proof of ordered paramodulation for Horn clauses with equality, where well-foundedness of the ordering suffices. Apart from the theoretical significance of this result, some potential applications motivating the interest of dropping the subterm property are given. The proof of the results included in this article, being still technical in some parts, is pretty much shorter and easier to read than the one we have in the preliminary version of this work presented at the CADE, 2002 conference (Bofill, and Rubio, 2002, CADE, Vol. 2392 of LNAI, pp. 456–470).

Algebraic and Coalgebraic Logic Corner

Venema, Y. — 2009-03-26

Algebras of Relations and Relevance Logic

Mikulas, S. — 2009-03-26

We prove that algebras of binary relations whose similarity type includes intersection, composition, converse negation and the identity constant form a non-finitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of relevant logic with respect to binary relations.

Proof Complexity of the Cut-free Calculus of Structures

Jerabek, E. — 2009-03-26

We investigate the proof complexity of analytic subsystems of the deep inference proof system SKSg (the calculus of structures). Exploiting the fact that the cut rule (i) of SKSg corresponds to the ¬-left rule in the sequent calculus, we establish that the ‘analytic'system KSg+c has essentially the same complexity as the monotone Gentzen calculus MLK. In particular, KSg+c quasipolynomially simulates SKSg, and admits polynomial-size proofs of some variants of the pigeonhole principle.

Nominal Algebra and the HSP Theorem

Gabbay, M. J. — 2009-03-26

Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work, it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as first-order logic, the -calculus or process calculi. Nominal algebra has a semantics in nominal sets (sets with a finitely supported permutation action); previous work proved soundness and completeness. The HSP theorem characterizes the class of models of an algebraic theory as a class closed under homomorphic images, subalgebras and products, and is a fundamental result of universal algebra. It is not obvious that nominal algebra should satisfy the HSP theorem: nominal algebra axioms are subject to so-called freshness conditions which give them some flavour of implication; nominal sets have significantly richer structure than the sets semantics traditionally used in universal algebra. The usual method of proof for the HSP theorem does not obviously transfer to the nominal algebra setting. In this article, we give the constructions which show that, after all, a ‘nominal’ version of the HSP theorem holds for nominal algebra; it corresponds to closure under homomorphic images, subalgebras, products and an atoms-abstraction construction specific to nominal-style semantics.

Fuzzy Logic Corner

Baaz, M., Metcalfe, G. — 2009-03-26

Fixpoint and While Temporal Query Languages

Bidoit, N., Objois, M. — 2009-03-26

The article investigates the expressive power of implicit temporal query languages. These languages are designed while assuming temporal data modelled as sequences of data. So far, two styles of implicit temporal query languages are known: t-while like languages are based on a temporal extension of while with left and right moves; µtl like languages are a fixpoint extension of tl. This article focusses on comparing the expressive power of t-while style languages and µtl languages and provides complementary results with respect to Abiteboul et al. (1999, J. Comput. System Sci.,58, 54–68). The main contribution is the proof of the equivalence of the three following temporal languages: the non-inflationary variant of µtl, the language t-while and more surprisingly the language t-fixpoint.

Updating Epistemic Logic Programs

Zhang, Y. — 2009-03-26

We consider the problem of updating non-monotonic knowledge bases represented by epistemic logic programs where disjunctive information and notions of knowledge and belief can be explicitly expressed. We propose a formulation for epistemic logic program update based on a principle called minimal change and maximal coherence. The central feature of our approach is that during an update or a sequence of updates, contradictory information is removed on a basis of minimal change under the semantics of epistemic logic programs and then coherent information is maximally retained in the update result. Through various update scenarios, we show that our approach provides both semantic and syntactic characterizations for an update problem. We also investigate essential semantic properties of epistemic logic program update.

Indifferent Sets

Figueira, S., Miller, J. S., Nies, A. — 2009-03-26

We define the notion of indifferent set with respect to a given class of {0,1}-sequences. Roughly, for a set A in the class, a set of natural numbers I is indifferent for A with respect to the class if it does not matter how we change A at the positions in I: the new sequence continues to be in the given class. We are especially interested in studying those sets that are indifferent with respect to classes containing different types of stochastic sequences. For the class of Martin-Löf random sequences, we show that every random sequence has an infinite indifferent set and that there is no universal indifferent set. We show that indifferent sets must be sparse, in fact sparse enough to decide the halting problem. We prove the existence of co-c.e. indifferent sets, including a co-c.e. set that is indifferent for every 2-random sequence with respect to the class of random sequences. For the class of absolutely normal numbers, we show that there are computable indifferent sets with respect to that class and we conclude that there is an absolutely normal real number in every non-trivial many-one degree.

Logic and Computation in the Real World: CiE 2007

Barry Cooper, S., Lowe, B., Sorbi, A. — 2009-01-22

K-Triviality of Closed Sets and Continuous Functions

Barmpalias, G., Cenzer, D., Remmel, J. B., Weber, R. — 2009-01-22

We investigate the notion of K-triviality for closed sets and continuous functions in 2^N. For every K-trivial degree d, there exists a closed set of degree d and a continuous function of degree d. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial $${\Pi }_{1}^{0}$$ class with no computable elements. A closed set is K-trivial if and only if it is the set of zeroes of some K-trivial continuous function. We give a density result for the Medvedev degrees of K-trivial $${\Pi }_{1}^{0}$$ sets. If W ≤_TA', then W can compute a path through every A'-decidable random closed set if and only if W _TA'.

RZ: a Tool for Bringing Constructive and Computable Mathematics Closer to Programming Practice

Bauer, A., Stone, C. A. — 2009-01-22

Realizability theory is not just a fundamental tool in logic and computability. It also has direct application to the design and implementation of programs, since it can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between the worlds of constructive mathematics and programming. By using the realizability interpretation of constructive mathematics, RZ translates specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools.

Borel Complexity of Topological Operations on Computable Metric Spaces

Brattka, V., Gherardi, G. — 2009-01-22

We study the Borel complexity of topological operations on closed subsets of computable metric spaces. The investigated operations include set theoretic operations such as union and intersection, but also typical topological operations such as the closure of the complement, the closure of the interior, the boundary and the derivative of a set. These operations are studied with respect to different computability structures on the hyperspace of closed subsets. These structures include positive or negative information on the represented closed subsets. Topologically, they correspond to the lower or upper Fell topology, respectively, and the induced computability concepts generalize the classical notions of recursively enumerable (r.e.) or co-r.e. subsets, respectively. The operations are classified with respect to effective measurability in the Borel hierarchy and it turns out that most operations can be located in the first three levels of the hierarchy, or they are not even Borel measurable at all. In some cases the effective Borel measurability depends on further properties of the underlying metric spaces, such as effective local compactness and effective local connectedness.

Pseudojumps and Formula Classes

Cenzer, D., Laforte, G., Wu, G. — 2009-01-22

For a pseudojump V^X and a $${\Pi }_{1}^{0}$$ class P, we consider properties of the set {V^X:X P}.We show that if P is Medvedev complete or if P has positive measure, and Ø' ≤_T C, then there exists X P with V^X _T C. We examine the consequences when V^X is Turing incomparable with V^Y for X != Y in P and when $${W}_{e}^{X}={W}_{e}^{Y}$$ for all X, Y P. Finally, we give a characterization of the jump in terms of $${\Pi }_{1}^{0}$$ classes.

Subexponential Time and Fixed-parameter Tractability: Exploiting the Miniaturization Mapping

Chen, Y., Flum, J. — 2009-01-22

Recently it has been shown that the miniaturization mapping M faithfully translates bexponential parameterized complexity into (unbounded) parameterized complexity. We determine the pre-images under M of various (classes of) problems. For many parameterized problems whose underlying classical problem is in NP we show that the pre-images coincide with natural reparameterizations that take into account the amount of non-determinism needed to solve them.

Existentially Closed Models and Conservation Results in Bounded Arithmetic

Cordon-Franco, A., Fernandez-Margarit, A., Lara-Martin, F. F. — 2009-01-22

We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories Sⁱ₂ and Tⁱ₂ and prove that they are ^b_i conservative over their inference rule counterparts, and ^b_i conservative over their parameter-free versions. A similar analysis of the ^b_i-replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.

The Settling Time Reducibility Ordering and {Delta}Formula Sets

Csima, B. F. — 2009-01-22

The settling time reducibility ordering gives an ordering on computably enumerable sets based on their enumerations. The <_st ordering is in fact an ordering on c.e. sets, since it is independent of the particular enumeration chosen. In this article, we show that it is not possible to extend this ordering in an approximation-independent way to ⁰₂ sets in general, or even to n-c.e. sets for any fixed n ≥ 3.

Enumeration Degrees and Enumerability of Familes

Kalimullin, I. — 2009-01-22

We study the enumerability of families relative to the enumeration degrees. It is shown that if a family of finite sets is e-reducible to every non-zero e-degree, then the family is computably enumerable (c.e). On the another hand, we will find a non-c.e. family which is e-reducible to all non-zero e-degree. This allows to construct a model, whose (extended) degree spectrum coincides with the non-zero e-degrees.

The Uniformity Principle for {Sigma}-definability

Korovina, M., Kudinov, O. — 2009-01-22

This article is an extended version of the paper published in Korovina and Kudinov (2007, Lecture Notes in Computer Science, Vol. 4497, pp. 416–425). The main goal of this research is to develop logical tools and techniques for effective reasoning about continuous data based on -definability. In this article we invent the Uniformity Principleand prove it for -definability over the real numbers extended by open predicates. Using the Uniformity Principle, we investigate different approaches to enrich the language of -formulas in such a way that simplifies reasoning about computable continuous data without enlarging the class of -definable sets. In order to do reasoning about computability of certain continuous data we have to pick up an appropriate language of a structure representing these continuous data. We formulate several major conditions how to do that in a right direction. We also employ the Uniformity Principleto argue that our logical approach is a good way for formalization of computable continuous data in logical terms.

Infinite Computations and a Hierarchy in {Delta}3 Reconsidered

Rovan, B., Steskal, L. — 2009-01-22

In this note, we reconsider the results in Rovan and Steskal (2007, Vol. 4497 of Lecture Notes in Computer Science, pp. 660–669, Springer) concerning TMDC (Display Turing Machines with Control) with Chomsky like control language. We shall show that, under the given assumptions, various degrees of the control complexity do not give rise to a hierarchy of language families, thus correcting an error in Rovan and Steskal (2007, Vol. 4497 of Lecture Notes in Computer Science, pp. 660–669, Springer).

Undecidability in Some Structures Related to Computation Theory

Selivanov, V. L. — 2009-01-22

We show that many of the so called discrete weak semilattices considered earlier in a series of author's publications have hereditary undecidable first-order theories. Since such structures appear naturally in some parts of computation theory, we obtain several new undecidability results. This applies e.g. to the structures of complete numberings, of m-degrees of index sets and of the Wadge degrees of k-partitions in the Baire space and -algebraic domains. Whenever possible, we try to determine also the exact degrees of undecidability of the theories under discussion.

A Jump Inversion Theorem for the Degree Spectra

Soskova, A. A., Soskov, I. N. — 2009-01-22

In the present article, we continue the study of the properties of the spectra of structures as sets of degrees initiated in [11]. Here, we consider the relationships between the spectra and the jump spectra. Our first result is that every jump spectrum is also a spectrum. The main result sounds like a Jump inversion theorem. Namely, we show that if a spectrum A is contained in the set of the jumps of the degrees in some spectrum B then there exists a spectrum C such that CB and A is equal to the set of the jumps of the degrees in C.

Logical and Complexity-theoretic Aspects of Models of Computation with Restricted Access to Arrays

Stewart, I. A. — 2009-01-22

We study a class of program schemes, NPSB, in which, aside from basic assignments, non-deterministic guessing and while loops, we have access to arrays; but where these arrays are binary write-once in that they are initialized to ‘zero’ and can only ever be set to ‘one’. We show, amongst other results, that: NPSB can be realized as a vectorized Lindström logic; there are problems accepted by program schemes of NPSB that are not definable in the bounded-variable infinitary logic $${L}_{\infty \omega }^{\omega }$$; all problems accepted by the program schemes of NPSB have asymptotic probability 1 and on ordered structures, NPSB captures the complexity class L^NP. We give equivalences (on the class of all finite structures) of the complexity-theoretic question ‘Does NP equal PSPACE?’, where the logics and classes of program schemes involved in the equivalent statements define or accept only problems with asymptotic probability 0 or 1 and so do not cover many computationally trivial problems. The class of program schemes NPSB is actually the union of an infinite hierarchy of classes of program schemes. Finally, when we amend the semantics of our program schemes slightly, we find that the classes of the resulting hierarchy capture the complexity classes ^p_i (where i≥2) of the Polynomial Hierarchy PH.