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Proofs Without Syntax
 Annals of Mathematics
"... [M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatori ..."
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[M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graphtheoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h: C → G(φ), where G(φ) is a graph associated with φ and C is a coloured graph. The main theorem is soundness and completeness: φ is true if and only if there exists a combinatorial proof h: C → G(φ). 1.
Towards Hilbert's 24th Problem: Combinatorial Proof Invariants
, 2006
"... Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for ..."
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Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand’s theorem.
Representing strategies for the connection calculus in rewriting logic
 Inst. für Informatik, Universität KoblenzLandau
, 2005
"... Abstract. Rewriting logic can be used to prototype systems for automated deduction. In this paper, we illustrate how this approach allows experiments with deduction strategies in a flexible and conceptually satisfying way. This is achieved by exploiting the reflective property of rewriting logic. By ..."
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Abstract. Rewriting logic can be used to prototype systems for automated deduction. In this paper, we illustrate how this approach allows experiments with deduction strategies in a flexible and conceptually satisfying way. This is achieved by exploiting the reflective property of rewriting logic. By specifying a theorem prover in this way one quickly obtains a readable, reliable and reasonably efficient system which can be used both as a platform for tactic experiments and as a basis for an optimized implementation. The approach is illustrated by specifying a calculus for the connection method in rewriting logic which clearly separates rules from tactics.