Published: 28th March 2013
DOI: 10.4204/EPTCS.113
ISSN: 2075-2180


Proceedings Seventh Workshop on
Logical and Semantic Frameworks, with Applications
Rio de Janeiro, Brazil, September 29-30, 2012

Edited by: Delia Kesner and Petrucio Viana

Invited Presentation: A very brief introduction to hybrid logic and hybrid-logical proof-theory
Torben Braüner
Invited Presentation: A framework for access control in distributed environments
Maribel Fernández
Invited Presentation: A proof-theoretical discussion on the mechanization of propositional logics
Edward Hermann Haeusler
Invited Presentation: A survey of classical realizability
Alexandre Miquel
Proof nets and the call-by-value lambda-calculus
Beniamino Accattoli
Symmetries in Modal Logics
Carlos Areces, Guillaume Hoffmann and Ezequiel Orbe
Elementary Deduction Problem for Locally Stable Theories with Normal Forms
Mauricio Ayala-Rincón, Maribel Fernández and Daniele Nantes-Sobrinho
Minimal lambda-theories by ultraproducts
Antonio Bucciarelli, Alberto Carraro and Antonino Salibra
The untyped stack calculus and Bohm's theorem
Alberto Carraro
The stack calculus
Alberto Carraro, Thomas Ehrhard and Antonino Salibra
A weak HOAS approach to the POPLmark Challenge
Alberto Ciaffaglione and Ivan Scagnetto
Sequent Calculi for the classical fragment of Bochvar and Halldén's Nonsense Logics
Marcelo E. Coniglio and María I. Corbalán
Non determinism through type isomorphism
Alejandro Díaz-Caro and Gilles Dowek
Formalizing the Confluence of Orthogonal Rewriting Systems
Ana Cristina Rocha Oliveira and Mauricio Ayala-Rincón
A Graph Calculus for Predicate Logic
Paulo A. S. Veloso and Sheila R. M. Veloso


The aim of the International Workshop on Logical and Semantic Frameworks, with Applications (LSFA) is to provide an informal forum to discuss theoretical and practical aspects of Formal Frameworks in Computer Science. LSFA is an annual meeting, earlier LSFA meetings were held in Belo Horizonte (2011), Natal (2010), Brasilia (2009), Salvador (2008), Ouro Preto (2007) and Natal (2006).

This document contains the proceedings of the Seventh International Workshop on Logical and Semantic Frameworks, with Applications, which was held on September 29 and 30, 2012, in Rio de Janeiro, Brazil. It contains 11 regular papers (9 long and 2 short) accepted for presentation at the meeting, as well as extended abstracts of invited talks by Torben Braüner (Roskilde University, Denmark), Maribel Fernàndez (King's College London, United Kingdom), Edward Hermann Haeusler (PUC-Rio, Brazil) and Alexandre Miquel (École Normale Supérieure de Lyon, France).

Each submission was reviewed by at least three referees, and an electronic Program Committee (PC) meeting was held using the EasyChair system.

We are very grateful to our invited speakers for kindly accepting to give talks at LSFA 2012. We would also like to thank all the PC members for their active participation to the electronic PC discussion.

Last but not least, thanks the organizing committee of LSFA 2012, Christiano Braga (UFF), Renata de Freitas (UFF), Luiz Carlos Pereira (PUC-Rio, UFRJ), and Maurício Ayala-Rincón (UnB)

21th of February, 2013 Delia Kesner (Université Paris-Diderot, France)
Petrucio Viana (Universidade Federal Fluminense, Brazil)

A very brief introduction to hybrid logic and hybrid-logical proof-theory

Torben Braüner (Roskilde University)

Hybrid logic

Hybrid logics are obtained by adding further expressive power to ordinary modal logic. The history of hybrid logics goes back to the philosopher Arthur Prior's work in the 1960s. The most basic hybrid logic is obtained by extending ordinary modal logic with nominals , which are propositional symbols of a new sort. In the Kripke semantics a nominal is interpreted in a restricted way such that it is true at exactly one point (where the points represent possible worlds, times, locations, epistemic states, states in a computer, or something else). If the points are given a temporal reading, this enables the formalization of natural language statements that are true at exactly one time, for example

which is true at the time five o'clock May 10th 2007, but false at all other times. Such statements cannot be formalized in ordinary modal logic, the reason being that ordinary propositional are not restricted to being true at exactly one point.

Most hybrid logics involve further additional machinery than nominals, the most important example being satisfaction operators . The motivation for adding satisfaction operators is to be able to formalize a statement being true at a particular time, possible world, or something else. Such statements cannot be formalized in ordinary modal logic. For example, we want to be able to formalize that the statement "it is raining'' is true at the time five o'clock May 10th 2007, that is, that This can be formalized by the formula @ar where the nominal a stands for "it is five o'clock May 10th 2007'' as above and where r is an ordinary propositional symbol that stands for "it is raining''. It is the part @a of the formula @ar that is called a satisfaction operator. In general, if a is a nominal and φ is an arbitrary formula, then a new formula @aφ can be built. The formula @aφ expresses that the formula φ is true at one particular point, namely the point to which the nominal a refers.

The addition of hybrid-logical machinery increases the expressive power, but often decidability is retained. Hybrid logics are closely related to description logics, which are a family of decidable logics used for knowledge representation in Artificial Intelligence. In description logics the points in a Kripke model represent individuals in the specification of an ontology. At present, significant research effort is put into exploring the borderline between decidable and undecidable logics, one major reason being that decidability is important for computational applications.

Proof theory

The subject of proof-theory is the notion of proof and formal systems for representing proofs. There are a number of different types of proof systems. Some of the most important types are natural deduction systems, Gentzen sequent systems, tableau systems, and axiom systems. They are motivated in different ways: Proof systems of the first three types are suitable for actual reasoning. Axiom systems are usually not meant for actual reasoning, but are of a more foundational interest. When a decidable logic is considered, Gentzen and tableau systems have the desirable feature of often giving rise to decision procedures in a very direct way, therefore Gentzen and tableau systems lend themselves toward computer implementations.

Hybrid-logical proof-theory

There is little consensus about proof-theory for ordinary modal logic, especially in connection with natural deduction systems, Gentzen sequent systems, and tableau systems. Many modal-logical proof systems lack important proof-theoretic properties and the relationships between proof systems for different modal logics are often unclear. It turns out that these deficiencies can be remedied by hybrid-logical proof-theory.

To be more precise, we can give a spectrum of different types of proof-systems (natural deduction, Gentzen, tableau, and axiom systems) for a spectrum of different versions of hybrid logic (propositional, first-order, intensional first-order, and intuitionistic). The natural deduction and Gentzen systems satisfy the requirements that such systems are expected to satisfy: The natural deduction system satisfies normalization where normal derivations satisfy a version of the subformula property. The Gentzen system is cut-free and also the Gentzen derivations satisfy a version of the subformula property. Moreover, tableau-based decision procedures can be given for decidable fragments of hybrid logic. All these systems can be motivated independently, but the fact that the systems can be given corroborates the point of view that hybrid logic and hybrid-logical proof-theory is a natural enterprise.

Besides satisfying these general requirements, hybrid-logical proof-theory furthermore satisfies the more concrete requirement that proof systems for wide classes of hybrid logics can be given in a uniform way, for example, natural deduction systems for a wide class of hybrid logics can be obtained in a uniform way by adding derivation rules as appropriate. This is simply not possible in connection with standard proof-theory for ordinary modal logic.


[1] T. Braűner. Hybrid Logic and Its Proof-Theory, volume 37 of Applied Logic Series. Springer, 2011. doi:10.1007/978-94-007-0002-4

A framework for access control in distributed environments

Maribel Fernández (King's College London)

Access control is a fundamental aspect of computer security; it aims at protecting resources from non-authorized users. The generalized use of access control in distributed computing environments has increased the need for high-level declarative languages that enable security administrators to specify and analyze a wide range of complex policies.

The first attempts to develop formal theories to define and validate security policies have used first-order theorem provers, purpose-built logics, or flow-analysis. More recently, rewriting techniques have been fruitfully exploited in the context of security protocols (e.g., [2]), information leakage (e.g., [8]), and access control (e.g., [10, 5]). Rewriting systems present many advantages: they have a well-studied theory, and programming languages and tools are available for fast prototyping and analysis (e.g. Maude [6]). Once the policies are defined, they can be integrated into a standard implementation, using for instance the weaving techniques described in [11].

Over the last few years, a variety of access control models have been developed, often motivated by particular applications. The unifying metamodel for access control proposed in [1] identifies a core set of principles of access control; all the standard models can then be derived as instances of the metamodel.

In [4], we use the same approach to define a metamodel of access control for distributed environments where each component of the system preserves its autonomy. The notion of distributed environment that we consider is related to the notion of federation developed in the context of database systems [9,7]. We propose a formal framework for modeling (and enforcing) global access control policies that takes into account the local policies specified and maintained by each member of the federation. In particular we ensure the coherence of a global access control decision w.r.t. local access control requirements by specifying in a tunable way how to integrate access authorizations resulting from the local policies.

We provide a rewrite-based operational semantics that deals in a uniform way with distributed systems where different access control policies are maintained locally. In this way, properties of access control policies can be proved in a modular way. In particular, we are interested in consistency and totality properties of policies. These properties guarantee that access requests will be treated as expected. A consistent access control policy specifies at most one answer for each access request (i.e., an access request cannot be both granted and denied). Totality guarantees that every access request will be given an answer. We show how consistency and totality properties of access control policies can be derived from standard properties of the rewrite framework we use.

The results reported here are based on preliminary work presented in [3, 4].


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[4] C. Bertolissi and M. Fernández. Rewrite specifications of access control policies in distributed environments. In Proc. of STM 2010: 6th Workshop on Security and Trust Management, Athens, Greece, 2010, number 6710 in Lecture Notes in Computer Science. Springer, 2011. doi:10.1007/978-3-642-22444-7_4
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[11] A. Santana de Oliveira, E. Ke Wang, C. Kirchner, and H. Kirchner. Weaving rewrite-based access control policies. In Proceedings of the 2007 ACM workshop on Formal methods in security engineering, FMSE 2007, Fairfax, VA, USA, November 2, 2007, pages 71-80. ACM, 2007. doi:10.1145/1314436.1314446

A proof-theoretical discussion on the mechanization of propositional logics

Edward Hermann Haeusler (PUC-Rio)

Classical (K) and Intuitionistic (I) propositional logics are well-known and very useful basic logics. On top of each of them many reasoning systems for AI and foundations of program constructions have been proposed. As a matter of terminology, given a logic L, let Taut(L) = { α : ∀ A ∈ Models(L), A ⊨L α } and Sat(L) = { α : ∃ A ∈ Models(L), A ⊨L α }. Minimal logic (M) is the logic obtained from I by removing the intuitionistic absurdity rule (from ⊢ ⊥ infer ⊢ B, for any B). Purely implicational minimal logic (M) is the fragment of minimal logic dealing only with the → logical constant. This work discusses the computational complexity of Sat(L) and Taut(L), for arbitrary propositional logics L, and how it is (possibly) hard to have an efficient mechanization of their respective proof procedures.

Sat(I) and Taut(I) are known to be PSPACE-complete, while Taut(K) is CONP-complete and Sat(K) is NP-complete. The proofs of these facts were obtained by model-theoretic arguments on the respective logics. For example, Statman [6] provided a polynomial reduction of quantified boolean logic (Taut(QBL)) to Taut(I) that is based on the truth-value semantics of the former. Additionally, in the same article, he provided a proof-theoretical argument for a polynomial reduction of I to M. After his article, the computational complexities of Sat(M) and Taut(M) are also known to be PSPACE-complete, since COPSPACE = PSPACE. It is worth noting that in spite of how Statman's reduction of QBF to I is based on a truth-valued semantics he also uses a proof-theoretical result: Glyvenko's theorem (from Γ ⊢K α it follows Γ ⊢I ¬ ¬ α).

There are two representative ways of normalizing Natural Deduction derivations. Prawitz strategy transforms arbitrary ⊥-classical applications in the derivation into atomic ⊥-classical applications. It works for the { ∧, →, ¬ } fragment. On the other hand, Seldin strategy transforms the derivation into a new derivation with at most one ⊥-classical application, the last rule application in it. Both strategies are concluded by usual reductions for eliminating maximal (detour) formulas as in the intuitionistic case, since classical ⊥ rules do not yield maximal formulas. In [5] it is shown how these usual strategies to normalize classical derivations are strongly related to double negation translation from classical to intuitionistic logic. Prawitz and Seldin strategy are related to Gödel and Kolmogorov translations respectively. In fact Seldin strategy can be seen as an alternative proof of Glyvenko's theorem. It is worth noting that both translations are polynomial on the size of the original derivation.

From facts and results mentioned above, we observe that the sub-formula principle property holding in classical, intuitionistic and minimal logics is what allows one to polynomially relate derivations among these logics, performing a worst case analysis for Taut(I), Taut(M) and Taut(M). In this work, we extend the polynomial reduction of purely implicational logic to any Natural Deduction presented propositional logic that satisfies the sub-formula principle. More precisely we prove:

Theorem 1 Let L be a propositional logic that has a finitary, complete and sound Natural Deduction system satisfying the sub-formula principle. Then Taut(L) is PSPACE-hard.

We remark that some well-known logics, such as Propositional Dynamic Logic, PDL, and S5Cn (n ≥ 2), for example, hardly satisfy the sub-formula property, unless EXPTIME = PSPACE. A feasible proving systems for these logics hardly exists. These logics are not polynomially bounded either, unless EXPTIME = NP. Any logic satisfying sub-formula principle, according Theorem 1 cannot be polynomially bounded either, unless NP = PSPACE. It is be generally accepted that a mechanizable logic should only have polynomially bounded proofs. A logic weaker than Taut(M) might be polynomially bounded, but its practical use must be more restricted in face of the already discussed facts. An alternative position for considering a mechanizable logic is taken in [1]. A propositional proof system is called automatizable whenever it only produces proofs in time polynomially bounded by the size of the smallest proof of the same conclusion. When one consider these smallest proofs written in stronger systems than the original one, it is called weak automatizable. Under certain conditions, weak automatizability is equivalent to feasible interpolation property (see [1]). A proof system has feasible interpolation property whenever for every proof of α → β there is a Craig interpolant γ that is polynomially bounded on the size of the proof of α → β. Interpolants are useful lemmas that can be used to produce smaller proofs. There are examples showing that an exponential gap may exist between proofs without lemmas and with lemmas. Since Taut(M) satisfies the subformula principle, Craig interpolation property must hold. However, interpolants are, in most of the cases, either the conclusion β or the premise α, and although they are polynomially bounded on the size of the proof they do not help producing smaller proofs. Despite Taut(M) satisfying feasible interpolation property, this does not provide any clue on the definition of an efficient propositional proof system for it.

We conclude this work by commenting some possible heuristics to provide short (polynomially bounded) proofs. Introduction of interpolants, or lemmata, is the most well-known heuristic. In [2] and [3] we show that it does not work at all and we propose the use of substitution/unification together with graph-based presentation in order to have short proofs. An example inspired in the Fibonnaci series is shown. This example is exponentially sized, it does not have any non-trivial and short interpolant and a proof polynomially bounded can be obtained by the application of the proposed heuristics.

A previous version of this work was presented at [4].


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A survey of classical realizability

Alexandre Miquel (ENS-Lyon)

The theory of classical realizability was introduced by Krivine [8] in the middle of the 90's in order to analyze the computational contents of classical proofs, following the connection between classical reasoning and control operators discovered by Griffin [4]. More than an extension of the theory of intuitionistic realizability, classical realizability is a complete reformulation of the very principles of realizability based on a combination [12, 11] of Kleene's realizability [5] with Friedman's A-translation [3].

One of the most interesting aspects of the theory is that it is highly modular: although it was originally developed for second order arithmetic, classical realizability naturally extends to more expressive frameworks such as Zermelo-Fraenkel set theory [6] or the calculus of constructions with universes [10]. Moreover, the underlying language of realizers can be freely enriched with new instructions in order to realize extra reasoning principles, such as the axiom of dependent choices [7]. More recently, Krivine [9] proposed an ambitious research program, whose aim is to unify the methods of classical realizability with Cohen's forcing [1, 2] in order to extract programs from proofs obtained by the forcing technique.

In this talk, I shall survey the methods and the main results of classical realizability, while presenting some of its specific problems, such as the specification problem. I shall also discuss the perspectives of combining classical realizability with the technique of forcing, showing that this combination is not only interesting from a pure model theoretic point of view, but that it can also bring new insights about a possible widening of the proofs-as-programs correspondence beyond the limits of pure functional programming.


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