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Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity
 LOGIC JOURNAL OF THE IGPL
, 2003
"... Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessi ..."
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Cited by 7 (5 self)
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Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural twosorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result
Polynomial Size DeepInference Proofs Instead Of Exponential Size ShallowInference Proofs
, 2004
"... ..."
Turning Cycles into Spirals
, 1999
"... Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14 ..."
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Cited by 6 (3 self)
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Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14]. There is a link between the presence of cut formulas with nested quantifiers and the nonelementary expansion needed to prove a theorem without the help of such formulas. If one considers the graph defined by tracing the flow of occurrences of formulas (in the sense of [2]) for proofs allowing a nonelementary compression, one Preprint submitted to Elsevier Preprint 7 November 1997 finds that such graphs contain cycles [5] or almost cyclic structures[6]. These cycles codify in a small space (i.e. a proof with a small number of lines) all the information which is present in the proof once cuts on formulas wit
Some notes on proofs with alpha graphs
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... It is wellknown that Peirce’s Alpha graphs correspond to propositional logic (PL). Nonetheless, Peirce’s calculus for Alpha graphs differs to a large extent to the common calculi for PL. In this paper, some aspects of Peirce’s calculus are exploited. First of all, it is shown that the erasurerule ..."
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Cited by 5 (3 self)
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It is wellknown that Peirce’s Alpha graphs correspond to propositional logic (PL). Nonetheless, Peirce’s calculus for Alpha graphs differs to a large extent to the common calculi for PL. In this paper, some aspects of Peirce’s calculus are exploited. First of all, it is shown that the erasurerule of Peirce’s calculus, which is the only rule which does not enjoy the finite choice property, is admissible. Then it is shown that this calculus is faster than the common cutfree calculi for propositional logic by providing formal derivations with polynomial lengths of Statman’s formulas. Finally a natural generalization of Peirce’s calculus (including the erasurerule) is provided such that we can find proofs linear in the number of propositional variables used in the formular, depending on the number of propositional variables in the formula.
Separating daglike and treelike proof systems
 Accepted in LICS
, 2007
"... We show that treelike (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cutfree PK. Generally, we exhibit a family of sequents that have polynomial size cutfree proofs but requires superpolynomial treelike proofs even when the cut rule is allowed on ..."
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Cited by 4 (1 self)
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We show that treelike (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cutfree PK. Generally, we exhibit a family of sequents that have polynomial size cutfree proofs but requires superpolynomial treelike proofs even when the cut rule is allowed on a class of cutformulas that satisfies some plausible hardness assumption. This gives (in some cases, conditional) negative answers to several questions from a recent work of Maciel and Pitassi (LICS 2006). Our technique is inspired by the technique from Maciel and Pitassi. While the sequents used in earlier work are derived from the Pigeonhole principle, here we generalize Statman’s sequents. This gives the desired separation, and at the same time provides stronger results in some cases. 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.
Interaction and Depth against Nondeterminism in Deep Inference Proof Search
, 2007
"... Deep inference is a proof theoretical methodology that generalises the traditional notion of inference of the sequent calculus. Deep inference provides more freedom in design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of expone ..."
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Cited by 3 (1 self)
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Deep inference is a proof theoretical methodology that generalises the traditional notion of inference of the sequent calculus. Deep inference provides more freedom in design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of exponentially shorter analytic proofs becomes possible, but with the cost of a greater nondeterminism than in the sequent calculus. In this paper, we extend our previous work on proof search with deep inference deductive systems. We argue that, by exploiting an interaction and depth scheme in the logical expressions, the nondeterminism in proof search can be reduced without losing the shorter proofs and without sacrificing from proof theoretical cleanliness.
The Number of Proof Lines and the Size of Proofs. in First Order Logic
"... There are two basic ways or measuring the complexity (or length) or proors: (1) to count the number or proor lines, (2) to count the total size or the proor (i.e. to count each symbol). Trivially the size is an upper bound to the number or proor lines. It is much more difficult to bound the size usi ..."
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Cited by 2 (0 self)
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There are two basic ways or measuring the complexity (or length) or proors: (1) to count the number or proor lines, (2) to count the total size or the proor (i.e. to count each symbol). Trivially the size is an upper bound to the number or proor lines. It is much more difficult to bound the size using the number or proor lines. ff we consider logic without runction symbols a reasonable bound can be proved (see Proposition 3.4). ff runction symbols are allowed, then the situation is considerably more complicated. In such a case rormulas in the proor may contain large terms and it is difficult to find some bounds to the size or these terms using only the information about the number or proor lines. There are still important open problems bere which show that the role or terms in the first order logic is not quite well understood. Some papers about this subject are rather difficult to read, Dne reason being that they consider general classes or logical calculi: Thererore we decided to consider just Dne particular calculus, Gentzen's wellknown calculus LK as presented in [TJ. Our results generalize trivially to theories given by a finite set or