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Reflections on Symmetry
 Complementizers and WH Constructions
, 2002
"... Abstract. Whilst it is generally accepted as a positive criterion, affordance only gives the weakest of hints for interactive systems designers. This paper shows how useful it is to consider affordance as generated by a correspondence between program symmetries and user interface symmetries. Symmetr ..."
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Cited by 8 (4 self)
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Abstract. Whilst it is generally accepted as a positive criterion, affordance only gives the weakest of hints for interactive systems designers. This paper shows how useful it is to consider affordance as generated by a correspondence between program symmetries and user interface symmetries. Symmetries in state spaces (for instance, as might be visualised in statecharts) can be carried through to user interfaces and into user manuals, with beneficial results. Exploiting affordances, understood in this way, in addition to their well known user interface benefits, makes programs simpler and more reliable, and makes user manuals shorter. 1
The Cost of a Cycle is a Square
, 1999
"... The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be nonelementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an ..."
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Cited by 3 (2 self)
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The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be nonelementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with O(k n+1 ) lines. In particular, there is a quadratic time algorithm which eliminates a single cycle from a proof. These results are motivated by the search for general methods on proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search.
Streams and Strings in Formal Proofs
"... Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. ..."
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Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. In our approach, "logic" is often forgotten and combinatorial properties of graphs are taken into account to explain logical phenomena.
Complexity and ordinary life, and some mathematics in both
, 2003
"... What is mathematics, exactly? This is a somewhat complicated question, with no simple answer. In any event, mathematics is like a large place, with many regions and villages, and many different ways of doing things. One can also try to make up new ways of doing things, in connection with whatever mi ..."
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What is mathematics, exactly? This is a somewhat complicated question, with no simple answer. In any event, mathematics is like a large place, with many regions and villages, and many different ways of doing things. One can also try to make up new ways of doing things, in connection with whatever might be of interest. Let us look here at a few points which can come up naturally in ordinary life, as well as involve some substantial mathematics (in some of their forms). Differences between answers of “yes ” and “no” Imagine the following kind of question: “Does soandso have a pencil in his or her office?” If someone finds a pencil in the office, then that provides a way in which an answer of “yes ” can be clearly established. The pencil can simply be shown. An answer of “no ” is quite different, and apparently more complicated. How can one establish an answer of “no”, without just going through all of the contents of the office?
Undecidability of the Problem of Ends for Automatic Graphs and Graph Substitution Systems
, 1999
"... Here is shown that the problem consisting in deciding whether an innite automatic graph has more than one end is recursively undecidable. The proof uses variations of the notions of selfstabilizing machine and of Turing machine configuration graph. A connection is established between automatic graph ..."
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Here is shown that the problem consisting in deciding whether an innite automatic graph has more than one end is recursively undecidable. The proof uses variations of the notions of selfstabilizing machine and of Turing machine configuration graph. A connection is established between automatic graphs and graph substitution systems. It is shown how to encode a sequence of finite graphs obtained by iterating some graph substitution by the spheres of an automatic graph and reciprocally. The question of the simultaneous connectivity of all the graphs of such sequences is then proved to be undecidable.
Alessandra Carbone Group Cancellation and Resolution
"... Abstract. We establish a connection between the geometric methods developed in the combinatorial theory of small cancellation and the propositional resolution calculus. We define a precise correspondence between resolution proofs in logic and diagrams in small cancellation theory, and as a consequen ..."
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Abstract. We establish a connection between the geometric methods developed in the combinatorial theory of small cancellation and the propositional resolution calculus. We define a precise correspondence between resolution proofs in logic and diagrams in small cancellation theory, and as a consequence, we derive that a resolution proof is a 2dimensional process. The isoperimetric function defined on diagrams corresponds to the length of resolution proofs.
Pathways of deduction A. Carbone
"... Cyclic structures underlie formal mathematical reasoning, and replication and folding play a crucial role in the complexity of proofs. These two aspects of the geometry of proofs are discussed. 1 Deductions, foldings and the brain Different models of various regions of the brain have been proposed a ..."
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Cyclic structures underlie formal mathematical reasoning, and replication and folding play a crucial role in the complexity of proofs. These two aspects of the geometry of proofs are discussed. 1 Deductions, foldings and the brain Different models of various regions of the brain have been proposed and they stimulated the discussion on the way our mind works. The essential feature of most of these models is the hierarchical structure which is underlying the organization. What we “see ” is nevertheless not necessarily the basic mechanism. Recent studies in computational complexity and proof theory reveal that hierarchical organizations, even though structurally appealing, are computationally inefficient. In fact, our brain seems to be “fast ” in performing certain tasks (such as perceiving the presence of an animal in the landscape, or intuitively grasping a complicated mathematical idea) and extremely “slow ” in performing others (as the construction of a mathematical
Departamento de Ciência da Computa»c~ao,
"... One of the most important open research problems in computer science nowadays is the \P=NP? " question [5].The answer to this question corresponds to knowing if decision problems that can be solved by a polynomialtime nondeterministic algorithm can also be solved by polynomialtime determinist ..."
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One of the most important open research problems in computer science nowadays is the \P=NP? " question [5].The answer to this question corresponds to knowing if decision problems that can be solved by a polynomialtime nondeterministic algorithm can also be solved by polynomialtime deterministic algorithm. The inference rules of automated deduction systems are typically nondeterministic in nature. Thus, in order to obtain a mechanical procedure, inference rules need to be complemented by another component, usually called strategy or search plan, which is responsible for the control of the inference rules [2]. In this paper we present the design and implementation of a multistrategy theorem prover based on the KE Tableau System [9]. A multistrategy theorem prover is a theorem prover where we can vary the strategy without modifying the core of the implementation. It can be used for three purposes: educational, exploratory and adaptive. For educational purposes, it can be used to illustrate how the choice of a strategy can a®ect performance of the prover. As an exploratory tool, a multistrategy theorem prover can be used to test new strategies and compare them with others. And we can also think of an adaptive multistrategy theorem prover that changes the strategy used according to features of the problem presented to it. To achieve the goal of constructing a welldesigned and e±cient multistrategy theorem prover, we are using a new software development method, aspect orientation [12], that allows a better modularization of crosscutting concerns such as strategies. The aspectoriented MultiStrategy Tableau Prover whose implementation we present here obtained excellent results compared with a similar tableaubased theorem prover [11]. 1