On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics

We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search

Detecting sequential structure

Programming by demonstration requires detection and analysis of sequential patterns in a user’s input, and the synthesis of an appropriate structural model that can be used for prediction. This paper describes SEQUITUR, a scheme for inducing a structural description of a sequence from a single example. SEQUITUR integrates several different inference techniques: identification of lexical subsequences or vocabulary elements, hierarchical structuring of such subsequences, identification of elements that have equivalent usage patterns, inference of programming constructs such as looping and branching, generalisation by unifying grammar rules, and the detection of procedural substructure., Although SEQUITUR operates with abstract sequences, a number of concrete illustrations are provided

On the correspondence between display postulates and deep inference in nested sequent calculi for tense logics

We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search

Almost structural completeness; an algebraic approach

A deductive system is structurally complete if its admissible inference rules are derivable. For several important systems, like modal logic S5, failure of structural completeness is caused only by the underivability of passive rules, i.e. rules that can not be applied to theorems of the system. Neglecting passive rules leads to the notion of almost structural completeness, that means, derivablity of admissible non-passive rules. Almost structural completeness for quasivarieties and varieties of general algebras is investigated here by purely algebraic means. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: expressed with finitely presented algebras, and with subdirectly irreducible algebras. One is restricted to quasivarieties with finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Secondly, examples of almost structurally complete varieties are provided Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of S4 modal logic. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification in it is not unitary. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras

The charm of structural neuroimaging in insanity evaluations. guidelines to avoid misinterpretation of the findings

Despite the popularity of structural neuroimaging techniques in twenty-first-century research, its results have had limited translational impact in real-world settings, where inferences need to be made at the individual level. Structural neuroimaging methods are now introduced frequently to aid in assessing defendants for insanity in criminal forensic evaluations, with the aim of providing “convergence” of evidence on the mens rea of the defendant. This approach may provide pivotal support for judges’ decisions. Although neuroimaging aims to reduce uncertainty and controversies in legal settings and to increase the objectivity of criminal rulings, the application of structural neuroimaging in forensic settings is hampered by cognitive biases in the evaluation of evidence that lead to misinterpretation of the imaging results. It is thus increasingly important to have clear guidelines on the correct ways to apply and interpret neuroimaging evidence. In the current paper, we review the literature concerning structural neuroimaging in court settings with the aim of identifying rules for its correct application and interpretation. These rules, which aim to decrease the risk of biases, focus on the importance of (i) descriptive diagnoses, (ii) anatomo-clinical correlation, (iii) brain plasticity and (iv) avoiding logical fallacies, such as reverse inference. In addition, through the analysis of real forensic cases, we describe errors frequently observed due to incorrect interpretations of imaging. Clear guidelines for both the correct circumstances for introducing neuroimaging and its eventual interpretation are defined

Focused labeled proof systems for modal logic

International audienceFocused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules from Gentzen's original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system G3K of Negri for the modal logic K and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in G3K corresponds to a bipole—a pair of a positive and a negative phases—in LKF. Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs

Type Classes for Lightweight Substructural Types

Linear and substructural types are powerful tools, but adding them to standard functional programming languages often means introducing extra annotations and typing machinery. We propose a lightweight substructural type system design that recasts the structural rules of weakening and contraction as type classes; we demonstrate this design in a prototype language, Clamp. Clamp supports polymorphic substructural types as well as an expressive system of mutable references. At the same time, it adds little additional overhead to a standard Damas-Hindley-Milner type system enriched with type classes. We have established type safety for the core model and implemented a type checker with type inference in Haskell.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441