Schemata

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Schemata as Building Blocks: Does Size Matter?

We analyze the schema theorem and the building block hypothesis using a recently derived, exact schemata evolution equation. We derive a new schema theorem based on the concept of effective fitness showing that schemata of higher than average effective fitness receive an exponentially increasing number of trials over time. The building block hypothesis is a natural consequence in that the equation shows how fit schemata are constructed from fit sub-schemata. However, we show that generically there is no preference for short, low-order schemata. In the case where schema reconstruction is favoured over schema destruction large schemata tend to be favoured. As a corollary of the evolution equation we prove Geiringer's theorem. We give supporting numerical evidence for our claims in both non-epsitatic and epistatic landscapes.Comment: 17 pages, 10 postscript figure

Linear Temporal Logic and Propositional Schemata, Back and Forth (extended version)

This paper relates the well-known Linear Temporal Logic with the logic of propositional schemata introduced by the authors. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one logic to the other. Some consequences about complexity are given. We report about first experiments and the consequences about possible improvements in existing implementations are analyzed.Comment: Extended version of a paper submitted at TIME 2011: contains proofs, additional examples & figures, additional comparison between classical LTL/schemata algorithms up to the provided translations, and an example of how to do model checking with schemata; 36 pages, 8 figure

The role of affect and cognitive schemata in the assessment of psychopathy

This thesis examined psychopathy, cognitive schemata and affect in forensic and community populations. This was to identify whether cognitive schemata and affect would assist in the assessment of psychopathy. Study one was conducted on 38 male high secure hospital patients and 38 male prisoners. It focused on the assessment of psychopathy and cognitive schemata. It was predicted that psychopathy would be positively related to negative schemata and early maladaptive schemata and negatively related to positive schemata. This prediction was supported with the exception of Early Maladaptive Schemata. Study two was conducted on 38 male high secure hospital patients and 38 male prisoners and also examined psychopathy and affect. It further explored positive schemata that was significant in study one. It was predicted that psychopathy would be positively related to errors on affective word sentence completion with slower response times. These predictions were not supported. The third study included 101 male prisoners and 108 male university students. An assessment of cognitive schema and affect was also developed. A further core prediction was that psychopathy would have a positive relationship with detached affect and results supported this. Contrary to prediction, it was found that psychopathy was higher in the student group compared to the prisoner group. Study four further explored the core predictions and included an examination of psychopathy, cognitive schema, affect and the 'Big Five' in 174 prisoners and 200 male students. The predictions were supported that psychopathy would be negatively related to positive cognitive schemata and positively related to negative cognitive schemata, in both groups. The predictions that detached affect would be significant to psychopathy was again supported. Contrary to prediction psychopathy was found to be higher in the student group. The current research indicates that cognitive schemata and affect are related to psychopathy. It also shows that similar cognitive profiles of psychopathy are demonstrated in prison and student groups that relate to affect. Further, it highlights the neglected role of positive schemata in psychopathy. Future research could consider the role of positive schemata and refine the cognitive profile in psychopathy, it could also examine the newly proposed cognitive behavioural model of psychopathy

Reasoning on Schemata of Formulae

A logic is presented for reasoning on iterated sequences of formulae over some given base language. The considered sequences, or "schemata", are defined inductively, on some algebraic structure (for instance the natural numbers, the lists, the trees etc.). A proof procedure is proposed to relate the satisfiability problem for schemata to that of finite disjunctions of base formulae. It is shown that this procedure is sound, complete and terminating, hence the basic computational properties of the base language can be carried over to schemata

A Decidable Class of Nested Iterated Schemata (extended version)

Many problems can be specified by patterns of propositional formulae depending on a parameter, e.g. the specification of a circuit usually depends on the number of bits of its input. We define a logic whose formulae, called "iterated schemata", allow to express such patterns. Schemata extend propositional logic with indexed propositions, e.g. P_i, P_i+1, P_1, and with generalized connectives, e.g. /\i=1..n or i=1..n (called "iterations") where n is an (unbound) integer variable called a "parameter". The expressive power of iterated schemata is strictly greater than propositional logic: it is even out of the scope of first-order logic. We define a proof procedure, called DPLL*, that can prove that a schema is satisfiable for at least one value of its parameter, in the spirit of the DPLL procedure. However the converse problem, i.e. proving that a schema is unsatisfiable for every value of the parameter, is undecidable so DPLL* does not terminate in general. Still, we prove that it terminates for schemata of a syntactic subclass called "regularly nested". This is the first non trivial class for which DPLL* is proved to terminate. Furthermore the class of regularly nested schemata is the first decidable class to allow nesting of iterations, i.e. to allow schemata of the form /\i=1..n (/\j=1..n ...).Comment: 43 pages, extended version of "A Decidable Class of Nested Iterated Schemata", submitted to IJCAR 200

Generating Schemata of Resolution Proofs

Two distinct algorithms are presented to extract (schemata of) resolution proofs from closed tableaux for propositional schemata. The first one handles the most efficient version of the tableau calculus but generates very complex derivations (denoted by rather elaborate rewrite systems). The second one has the advantage that much simpler systems can be obtained, however the considered proof procedure is less efficient

Learning-based constraints on schemata

Schemata are frequently used in cognitive science as a descriptive framework for explaining the units of knowledge. However, the specific properties which comprise a schema are not consistent across authors. In this paper we attempt to ground the concept of a schema based on constraints arising from issues of learning. To do this, we consider the different forms of schemata used in computational models of learning. We propose a framework for comparing forms of schemata which is based on the underlying representation used by each model, and the mechanisms used for learning and retrieving information from its memory. Based on these three characteristics, we compare examples from three classes of model, identified by their underlying representations, specifically: neural network, production-rule and symbolic network models

Integrating a Global Induction Mechanism into a Sequent Calculus

Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof theoretic properties of the calculus. To the best of our knowledge, there are no such calculi which allow cut-elimination to a normal form with the subformula property, i.e. every formula occurring in the proof is a subformula of the end sequent. Proof schemata are a variant of LK-proofs able to simulate induction by linking proofs together. There exists a schematic normal form which has comparable proof theoretic behaviour to normal forms with the subformula property. However, a calculus for the construction of proof schemata does not exist. In this paper, we introduce a calculus for proof schemata and prove soundness and completeness with respect to a fragment of the inductive arguments formalizable in Peano arithmetic.Comment: 16 page