[pt] Esta tese apresenta um novo cálculo de sequente, correto e completo para a Lógica Proposicional Minimal Implicacional (M →). LMT → destina-se a ser usado para a busca de provas em M →, em uma abordagem bottom-up. A Terminação do cálculo é garantida por uma estratégia de aplicação de regras que força uma maneira ordenada no procedimento de busca de provas de tal forma que todas as combinações possíveis são exploradas. Para uma fórmula inicial α, as provas em LMT→ têm um limite superior de |α|.2 |α|+1+2·log2|α|, que juntamente com a estratégia do sistema, garantem a decidibilidade do mesmo. As regras do sistema são concebidas para lidar com a necessidade de repetição de hipóteses e a natureza de perda de contexto da regra → esquerda , evitando a ocorrência de loops e o uso de backtracking. Portanto, a busca de prova em LMT → é determinística, sempre executando buscas no sentido forward. LMT → tem a propriedade de permitir a extração de contramodelos a partir de buscas de prova que falharam (bicompletude), isto é, a árvore de tentativa de prova de um ramo totalmente expandido produz um modelo de Kripke que falsifica a fórmula inicial. A geração de contra-modelo (usando a semântica Kripke) é obtida como consequência da completude do sistema. LMT→ é implementado como um provador de teoremas interativo baseado no cálculo proposto aqui. Comparamos nosso cálculo com outros sistemas dedutivos conhecidos para M →, especialmente com Tableaux no estilo Fitting, um método que também tem a propriedade de ser bicompleto. Também propomos aqui uma tradução de LMT → para o verificador de prova Dedukti como uma forma de avaliar a correção da implementação que desenvolvemos, no que diz respeito à especificação do sistema, além de torná-lo mais fácil de comparar com outros sistemas existentes. / [en] This thesis presents a new sequent calculus called LMT→ that has the properties to be terminating, sound and complete for Propositional Implicational Minimal Logic (M →). LMT→ is aimed to be used for proof search in M →, in a bottom-up approach. Termination of the calculus is guaranteed by a strategy of rule application that forces an ordered way to search for proofs such that all possible combinations are stressed. For an initial formula α, proofs in LMT→ has an upper bound of |α|.2 |α|+1+2·log2|α|, which together with the system strategy ensure decidability. System rules are conceived to deal with the necessity of hypothesis repetition and the contextsplitting nature of → left, avoiding the occurrence of loops and the usage of backtracking. Therefore, LMT→ steers the proof search always in a forward, deterministic manner. LMT→ has the property to allow extractability of counter-models from failed proof searches (bicompleteness), i.e., the attempt proof tree of an expanded branch produces a Kripke model that falsifies the initial formula. Counter-model generation (using Kripke semantics) is achieved as a consequence of the completeness of the system. LMT→ is implemented as an interactive theorem prover based on the calculus proposed here. We compare our calculus with other known deductive systems for M →, especially with Fitting s Tableaux, a method that also has the bicompleteness property. We also proposed here a translation of LMT→ to the Dedukti proof checker as a way to evaluate the correctness of the implementation regarding the system specification and to make our system easier to compare to others.

In this thesis I study several deductive systems for linear logic, its fragments, and some noncommutative extensions. All systems will be designed within the calculus of structures, which is a proof theoretical formalism for specifying logical systems, in the tradition of Hilbert's formalism, natural deduction, and the sequent calculus. Systems in the calculus of structures are based on two simple principles: deep inference and top-down symmetry. Together they have remarkable consequences for the properties of the logical systems. For example, for linear logic it is possible to design a deductive system, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. I will also show an extension of multiplicative exponential linear logic by a noncommutative, self-dual connective which is not representable in the sequent calculus. All systems enjoy the cut elimination property. Moreover, this can be proved independently from the sequent calculus via techniques that are based on the new top-down symmetry. Furthermore, for all systems, I will present several decomposition theorems which constitute a new type of normal form for derivations.