Logic Journal of IGPL - recent issues

Extended semantics and inference for the Independent Choice Logic

Riguzzi, F. — Tue, 17 Nov 2009 07:48:50 PST

The Independent Choice Logic (ICL), proposed by Poole, is a language for expressing probabilistic information in logic programming that adopts a distribution semantics: an ICL theory defines a distribution over a set of normal logic programs. The probability of a query is then given by the sum of the probabilities of the programs where the query is true.

The ICL semantics requires the theory to be acyclic. This is a strong limitation that rules out many interesting programs. In this paper we present an extension of the ICL semantics that allows theories to be modularly acyclic.

Inference with ICL can be performed with the Ailog2 system that computes explanations to queries and then makes them mutually incompatible by means of an iterative algorithm.

We propose the system PICL (for Probabilistic inference with ICL) that computes the explanations to queries by means of a modification of SLDNF-resolution and then makes the explanations mutually incompatible by means of Binary Decision Diagrams.

PICL and Ailog2 are compared on problems that involve computing the probability of a connection between two nodes in biological graphs and in social networks. Moreover, they are also applied to three games of dice.

The problems considered are easily expressible in P-log, a probabilistic language based on Answer Set Programming. Therefore, the Plog system was also applied to the programs.

PICL was able to handle larger problems than Ailog2 and Plog. Moreover, it was the fastest of the three algorithms except for one case of one of dice games.

Modelling evolvable component systems: Part I: A logical framework

Barringer, H., Gabbay, D., Rydeheard, D. — Tue, 17 Nov 2009 07:48:50 PST

We develop a logical modelling approach to describe evolvable computational systems. In this account, evolvable systems are built hierarchically from components where each component may have an associated supervisory process. The supervisor's purpose is to monitor and possibly change its associated component. Evolutionary change may be determined purely internally from observations made by the supervisor or may be in response to external change. Supervisory processes may be present at any level in the component hierarchy allowing us to use evolutionary behaviour as an integral part of system design.

We model such systems in a revision-based first-order logical framework in which supervisors are modelled as theories which are at a logical meta-level to the theories of their components. This enables evolutionary change of the component to be induced by revision-based changes of the supervisor at the meta-level. In this way, the intervention required in evolutionary change is modelled purely logically.

The hierarchical component-based structure is fairly intricate so we present the basic ideas firstly in a simple setting, the well-known blocks world, before introducing tree-based structures to represent component hierarchies. We also introduce some techniques for establishing the behaviour of evolvable systems specified in this logical framework. The ideas and concepts are driven by example throughout. We conclude with a more substantial example, that of a simple model of an evolvable system of automated bank teller machines.

Explicit substitutions calculi with one step Eta-reduction decided explicitly

Ventura, D., Ayala-Rincon, M., Kamareddine, F. — Tue, 17 Nov 2009 07:48:50 PST

It has long been argued that the notion of substitution in the -calculus needs to be made explicit. This resulted in many calculi have been developed in which the computational steps of the substitution operation involved in β-contractions have been atomised. In contrast to the great variety of developments for making explicit formalisations of the Beta rule, less work has been done for giving explicit definitions of the conditional Eta rule. In this paper constructive Eta rules are proposed for both the - and the s_e-calculi of explicit substitutions. Our results can be summarised as follows: 1) we introduce constructive and explicit definitions of the Eta rule in the - and the s_e-calculi, 2) we prove that these definitions are correct and preserve basic properties such as subject reduction. In particular, we show that the explicit definitions of the eta rules coincide with the Eta rule for pure -terms and that moreover, their application is decidable in the sense that Eta redices are effectively detected (and contracted). The formalisation of these Eta rules involves the development of specific calculi for explicitly checking the condition of the proposed Eta rules while constructing the Eta contractum.

Independence-friendly cylindric set algebras

Mann, A. L. — Tue, 17 Nov 2009 07:48:50 PST

Independence-friendly logic (IF logic) is a conservative extension of first-order logic that has the same expressive power as existential second-order logic. We attempt to algebraize IF logic in the same spirit as cylindric algebra.

We define independence-friendly cylindric set algebras (IF algebras) and investigate to what extent they satisfy the axioms of cylindric algebra. We ask whether the equational theory of IF algebras is finitely axiomatizable, and prove two partial results. First, every IF algebra over a structure is an expansion of a Kleene algebra. Moreover, the class of such Kleene algebras generates the variety of all Kleene algebras. Second, every one-dimensional IF algebra over a structure is an expansion of a monadic Kleene algebra. However, the class of such monadic Kleene algebras does not generate the variety of all monadic Kleene algebras.¹

Neat reducts and amalgamation in retrospect, a survey of results and some methods Part II: Results on amalgamation

Madarasz, J., Ahmed, T. S. — Tue, 17 Nov 2009 07:48:50 PST

Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the (seemingly very innocent) fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new unpublished results (answering long-standing open problems in the field) concerning neat reducts and amalgamation are given. (Theorems 11, 13, 19 and 31-38 are such). Several counterexamples which convey the gist of techniques used in this area are presented two of which are new (Theorem 19, Theorem 38.) It is known that the algebraic notion of amalgamation in a class of algebras corresponds to the metalogical notion of interpolation in the corresponding logic. Answers to open question in the recent paper [31] concerning both amalgamation and interpolation are summarized in tabular form at the end of this paper.

This paper appears in two parts. The first part contains results on neat reducts. The present second part contains results relating the notion of neat embeddings to various amalgamation properties.¹

Erratum to "The Ricean Objection: An Analogue of Rice's Theorem for First-Order Theories" * Logic Journal of the IGPL, 16(6): 585-590(2008)

Oliveira, I. C., Carnielli, W. — Tue, 17 Nov 2009 07:48:50 PST

16th Workshop on Logic, Language, Information and Computation (WoLLIC 2009)

Tue, 17 Nov 2009 07:48:51 PST

Acknowledgements

Tue, 17 Nov 2009 07:48:51 PST

Preface

Ayala-Rincon, M., Haeusler, E. H. — Thu, 08 Oct 2009 09:05:47 PDT

On the convergence of reduction-based and model-based methods in proof theory

Dowek, G. — Thu, 08 Oct 2009 09:05:48 PDT

In the recent past, the reduction-based and the model-based methods to prove cut elimination have converged, so that they now appear just as two sides of the same coin. This paper details some of the steps of this transformation.

From light logics to type assignments: a case study

Gaboardi, M., Rocca, S. R. D. — Thu, 08 Oct 2009 09:05:48 PDT

Using Soft Linear Logic (SLL) as case study, we analyze a method for transforming a light logic into a type assignment system for the -calculus, inheriting the complexity properties of the logics. Namely the typing assures the strong normalization in a number of steps polynomial in the size of the term, and moreover all polynomial functions can be computed by -terms that can be typed in the system. The proposed method is general enough to be used also for other light logics.

Natural deduction for the finite least fixed point logic with an infinitary rule

Arruda, A. M., Martins, A. T. — Thu, 08 Oct 2009 09:05:49 PDT

The notion of the least fixed point of an operator is widely applied in computer science as, for instance, in the context of query languages for relational databases. Some extensions of first-order classical logic (FOL) with fixed point operators, as the least fixed point logic (LFP), were proposed to deal with problems related to the expressivity of FOL. LFP captures the complexity class PTIME over the class of finite ordered structures. The descriptive characterization of computational classes is a central issue within finite model theory (FMT). Trahtenbrot's theorem states that validity over finite models is not recursively enumerable, that is, completeness fails over finite models. This result is based on an underlying assumption that any deductive system is of finite nature. However, we can relax such assumption as done in the scope of proof theory for arithmetic. Motivated by Gödel incompleteness theorems, proof theory for arithmetic offer an example of a true mathematically meaningful principle non-derivable in first-order arithmetic. One way of presenting this proof is based on a definition of a proof system with an infinitary rule, the -rule, that establishes the consistency of first-order arithmetic through a proof-theoretical perspective. Inspired in this rule, here we will propose an infinitary natural deduction system, and a sequent calculus version, for FOL and LFP restricted to finite models, FOL_fin and LFP_fin, respectively, we will prove soundness and completeness for them and also normal form theorems for them. With this infinitary deductive system for LFP_fin, we aim to present a proof theory for a logic traditionally investigated within the scope of FMT. It opens up an alternative way of proving results already obtained within FMT and also new ones through a proof-theoretical perspective.

Using modal logics to express and check global graph properties

Benevides, M. R. F., Schechter, L. M. — Thu, 08 Oct 2009 09:05:49 PDT

Graphs are among the most frequently used structures in Computer Science. Some of the properties that must be checked in many applications are connectivity, acyclicity and the Eulerian and Hamiltonian properties. In this work, we analyze how we can express these four properties with modal logics. This involves two issues: whether each of the modal languages under consideration has enough expressive power to describe these properties and how complex (computationally) it is to use these logics to actually test whether a given graph has some desired property. First, we show that these properties are not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal µ-calculus. We then show that it is possible to express some of the above properties in a basic hybrid logic. Unfortunately, the Hamiltonian and Eulerian properties still cannot be efficiently checked. In a second attempt, we propose an extension of CTL* with nominals and show that the Hamiltonian property can be more efficiently checked in this logic than in the previous one. In a third attempt, we extend the basic hybrid logic with the operator and show that we can check the Hamiltonian property with optimal (NP) complexity in this logic. Finally, we tackle the Eulerian property in two different ways. First, we develop a generic method to express edge-related properties in hybrid logics and use it to express the Eulerian property. Second, we express a necessary and sufficient condition for the Eulerian property to hold using a graded modal logic.

Acknowledgements

Thu, 08 Oct 2009 09:05:49 PDT

Distance-based non-deterministic semantics for reasoning with uncertainty

Arieli, O., Zamansky, A. — Wed, 15 Jul 2009 03:52:05 PDT

Non-deterministic matrices, a natural generalization of many-valued matrices, are semantic structures in which the value assigned to a complex formula may be chosen non-deterministically from a given set of options. We show that by combining non-deterministic matrices and distance-based considerations, one obtains a family of logics that are useful for reasoning with uncertainty. These logics are a conservative extension of those that are obtained by standard (i.e., deterministic) distance-based semantics, and so usual distance-based methods (in the context of, e.g., belief revision, information integration, and social choice theory) are easily simulated within our framework.

We investigate the basic properties of the distance-preferential non-deterministic logics, consider their application for reasoning with incomplete and inconsistent information, and show the correspondence between some particular entailments in our framework and well-known problems like max-SAT.

Every computably enumerable random real is provably computably enumerable random

Calude, C. S., Hay, N. J. — Wed, 15 Jul 2009 03:52:05 PDT

We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that "a real is c.e. and random iff it is the halting probability of a universal prefix-free Turing machine" can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem.

Our positive result can be contrasted with the case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost all random finite strings are not provably random in PA.

We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA, b) there exists a universal machine U such that, based on U, PA cannot prove the randomness of its halting probability.

The paper also includes a sharper form of the Kraft-Chaitin Theorem, as well as a formal proof of this theorem written with the proof assistant Isabelle.

Relational approach for a logic for order of magnitude qualitative reasoning with negligibility, non-closeness and distance

Golinska-Pilarek, J., Munoz-Velasco, E. — Wed, 15 Jul 2009 03:52:05 PDT

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for order-of-magnitude reasoning and its dual tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object.

The category of MV-pairs

Di Nola, A., Holcapek, M., Jenca, G. — Wed, 15 Jul 2009 03:52:05 PDT

An MV-pair is a pair (B,G), where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain condition. Recently it was proved by one of the authors that for an MV-pair (B,G), ~_G is an effect-algebraic congruence and B/~_G is an MV-algebra. Moreover, every MV-algebra M can be represented by an MV-pair in this way.

In this paper we show that one can define a suitable category of MV-pairs in such a way that there exist a faithful functor from the category of MV-algebras to the aforementioned category and a functor in the reversed direction.

HpsUL is not the logic of pseudo-uninorms and their residua

Wang, S., Zhao, B. — Wed, 15 Jul 2009 03:52:05 PDT

This paper presents several results on the non-commutative fuzzy logic HpsUL, a Hilbert system whose corresponding algebraic semantics is the variety of bounded representable residuated lattices. In particular, we prove that HpsUL is not complete with respect to algebras based on the real unit interval, which answers the question posed by Metcalfe, Olivetti and Gabbay and shows that HpsUL is not the logic of pseudo-uninorms and their residua.

MSC2000: 03B52, 03G10.

A simplified embedding of E into monomodal K

French, R. — Wed, 15 Jul 2009 03:52:05 PDT

In this paper we will provide a modal-to-modal translational embedding of E into K, simplifying a similar result which is obtainable using a novel translation due to S.K. Thomason.

Neat reducts and amalgamation in retrospect, a survey of results and some methods Part I: Results on neat reducts

Madarasz, J., Ahmed, T. S. — Wed, 15 Jul 2009 03:52:05 PDT

Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the (seemingly very innocent) fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new unpublished results (answering long-standing open problems in the field) concerning neat reducts and amalgamation are given. (Theorems 11, 13, 19 and 31-38 are such). Several counterexamples which convey the gist of techniques used in this area are presented two of which are new (Theorem 19, Theorem 38). It is known that the algebraic notion of amalgamation in a class of algebras corresponds to the metalogical notion of interpolation in the corresponding logic. Answers to open question in the recent paper [54] concerning both amalgamation and interpolation are summarized in tabular form at the end of this paper.

This paper appears in two parts. The present first part contains results on neat reducts. The second part contains results relating the notion of neat embeddings to various amalgamation properties.¹

Acknowledgements

Wed, 15 Jul 2009 03:52:05 PDT