1,721,029 research outputs found
Cut-elimination, substitution and normalisation
No full textDate of Acceptance: 01/2015We present a proof (of the main parts of which there is a formal version, checked with the Isabelle proof assistant) that, for a G3-style calculus covering all of intuitionistic zero-order logic, with an associated term calculus, and with a particular strongly normalising and confluent system of cut-reduction rules, every reduction step has, as its natural deduction translation, a sequence of zero or more reduction steps (detour reductions, permutation reductions or simplifications). This complements and (we believe) clarifies earlier work by (e.g.) Zucker and Pottinger on a question raised in 1971 by Kreisel.Peer reviewe
Tools and techniques for formalising structural proof theory
No full textWhilst results from Structural Proof Theory can be couched in many formalisms, it is
the sequent calculus which is the most amenable of the formalisms to metamathematical
treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are
all cases of a proof completed in the literature. Two intermediate results can be used to
drastically reduce the amount of effort needed in proofs of Cut admissibility: Weakening
and Invertibility. Indeed, whereas there are proofs of Cut admissibility which do not use
Invertibility, Weakening is almost always necessary. Use of these results simply shifts the
bureaucracy, however; Weakening and Invertibility, whilst more easy to prove, are still not
trivial. We give a framework under which sequent calculi can be codified and analysed,
which then allows us to prove various results: for a calculus to admit Weakening and for a
rule to be invertible in a calculus. For the latter, even though many calculi are investigated,
the general condition is simple and easily verified. The results have been applied to G3ip,
G3cp, G3s, G3-LC and G4ip.
Invertibility is important in another respect; that of proof-search. Should all rules in
a calculus be invertible, then terminating root-first proof search gives a decision procedure
for formulae without the need for back-tracking. To this end, we present some results
about the manipulation of rule sets. It is shown that the transformations do not affect the
expressiveness of the calculus, yet may render more rules invertible. These results can guide
the design of efficient calculi.
When using interactive proof assistants, every case of a proof, however complex, must be
addressed and proved before one can declare the result formalised. To do this in a human readable
way adds a further layer of complexity; most proof assistants give output which is
only legible to a skilled user of that proof assistant. We give human-readable formalisations
of Cut admissibility for G3cp and G3ip, Contraction admissibility for G4ip and Craig's
Interpolation Theorem for G3i using the Isar vernacular of Isabelle. We also formalise the
new invertibility results, in part using the package for reasoning about first-order languages,
Nominal Isabelle. Examples are given showing the effectiveness of the formalisation. The
formal proof of invertibility using the new methods is drastically shorter than the traditional,
direct method
On the relationship between hypersequent calculi and labelled sequent calculi for intermediate logics with geometric Kripke semantics
No full textIn this thesis we examine the relationship between hypersequent and some types of labelled sequent calculi for a subset of intermediate logics—logics between intuitionistic (Int), and classical logics—that have geometric Kripke semantics, which we call Int∗/Geo.
We introduce a novel calculus for a fragment of first-order classical logic, which
we call partially-shielded formulae (or PSF for short), that is adequate for expressing
the semantic validity of formulae in Int∗/Geo, and apply techniques from correspondence theory to provide translations of hypersequents, simply labelled sequents and relational sequents (simply labelled sequents with relational formulae) into PSF. Using these translations, we show that hypersequents and simply labelled sequents for calculi in
Int∗/Geo share the same models. We also use these translations to justify various techniques that we introduce for translating simply labelled sequents into relational sequents and vice versa. In particular, we introduce a technique called "transitive unfolding" for translating relational sequents into simply labelled sequents (and by extension, hypersequents) which preserves linear models in Int∗/Geo.
We introduce syntactic translations between hypersequent calculi and simply labelled
sequent calculi. We apply these translations to a novel hypersequent framework HG3ipm∗
for some logics in Int∗/Geo to obtain a corresponding simply labelled sequent framework LG3ipm∗, and to an existing simply labelled calculus for Int from the literature to obtain a novel hypersequent calculus for Int.
We introduce methods for translating a simply labelled sequent calculus into a cor-
responding relational calculus, and apply these methods to LG3ipm∗ to obtain a novel
relational framework RG3ipm∗ that bears similarities to existing calculi from the literature. We use transitive unfolding to translate proofs in RG3ipm∗ into proofs in LG3ipm∗ and HG3ipm∗ with the communication rule, which corresponds to the semantic restriction to linear models
Proof search issues in some non-classical logics
Full text linkThis thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut- free LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1-1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called 'permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics; Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a 'permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut- elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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