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Extracting Herbrand Disjunctions by Functional Interpretation
"... Abstract. Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary firstorder predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PLproofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of ..."
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Abstract. Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary firstorder predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PLproofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of two main steps: first we extract a functional realizer, next we compute the βnormalform of the realizer from which the Herbrand terms can be read off. Even though the extraction is carried out in the extended language, the terms are ordinary PLterms. In contrast to approaches to Herbrand’s theorem based on cut elimination or εelimination this extraction technique is, except for the normalization step, of low polynomial complexity, fully modular and furthermore allows an analysis of the structure of the Herbrand terms, in the spirit of Kreisel ([13]), already prior to the normalization step. It is expected that the implementation of functional interpretation in Schwichtenberg’s MINLOG system can be adapted to yield an efficient Herbrandterm extraction tool. 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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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.
Term Induction
, 2001
"... In this thesis we study a formal first order system T (tind) in the standard language L, of Gentzen’s LK, see [Tak87]. T (tind) extends LK by the following valid firstorder inference rule (A is quantifierfree). Γ, A(a), Λ → ∆, A(s(a)), Θ Γ, A(0), Λ → ∆, A(s n (0)), Θ (tind) This rule is called ter ..."
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In this thesis we study a formal first order system T (tind) in the standard language L, of Gentzen’s LK, see [Tak87]. T (tind) extends LK by the following valid firstorder inference rule (A is quantifierfree). Γ, A(a), Λ → ∆, A(s(a)), Θ Γ, A(0), Λ → ∆, A(s n (0)), Θ (tind) This rule is called term induction, it derives a restricted term built from successor s and the constant 0. We call such terms numerals. To characterise the difference between T (tind) and pure logic, we employ proof theoretic methods. Firstly we establish a variant of Herbrand’s Theorem for T (tind). Let ∃¯xF (¯x) be a Σ1 formula; provable by Π. Then there exists a disjunction � N i1 · · · � N il M1(s i1 (0),..., s il(0)) ∨ · · · ∨ Mm(s i1 (0),..., s il(0)), denoted by H that is valid for some N ∈ IN, furthermore the Mi are instances of F (ā). In T (tind) it is not possible to bound the length of Herbrand disjunctions in terms of proof length and logical complexity of the endformula as usual. The main result is that we can bound the length of the {s, 0}matrix of the above disjunctions in this way.
Refined Complexity Analysis of Cut Elimination
"... Abstract. In [1, 2] Zhang shows how the complexity of cut elimination depends primarily on the nesting of quantifiers in cut formulas. By studying the role of contractions in cut elimination we can refine that analysis and show how the complexity depends on a combination of contractions and quantifi ..."
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Abstract. In [1, 2] Zhang shows how the complexity of cut elimination depends primarily on the nesting of quantifiers in cut formulas. By studying the role of contractions in cut elimination we can refine that analysis and show how the complexity depends on a combination of contractions and quantifier nesting. With the refined analysis the upper bound on cut elimination coincides with Statman’s lower bound. Every nonelementary growth example must display a combination of nesting of quantifiers and contractions similar to Statman’s lower bound example. The upper and lower bounds on cut elimination immediately translate into bounds on Herbrand’s theorem. Finally we discuss the role of quantifier alternations and show an elementary upper bound for the ∀−∧case (resp. ∃−∨case). 1
INTRODUCTION TO THE COMBINATORICS AND COMPLEXITY OF CUT ELIMINATION
"... Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted ..."
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Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted
Chapter 1 Checking proofs
"... Argumentative practice in mathematics evidently takes a number of shapes. An important part of understanding mathematical argumentation, putting aside its special subject matters (numbers, shapes, spaces, sets, functions, etc.), is that mathematical argument often tends toward formality, and it ofte ..."
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Argumentative practice in mathematics evidently takes a number of shapes. An important part of understanding mathematical argumentation, putting aside its special subject matters (numbers, shapes, spaces, sets, functions, etc.), is that mathematical argument often tends toward formality, and it often has superlative epistemic goals:
Upper Bound on the Height of Terms in Proofs with Cuts
, 1998
"... We describe an upper bound on the heights of terms occurring in a most general unifier of a system of pairs of terms that contains unknowns of two types. An unknown belongs to the first type if all occurrences of this unknown have the same depth; we call such unknown an unknown of the cut type. ..."
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We describe an upper bound on the heights of terms occurring in a most general unifier of a system of pairs of terms that contains unknowns of two types. An unknown belongs to the first type if all occurrences of this unknown have the same depth; we call such unknown an unknown of the cut type. Unknowns of the second type (unknowns of not the cut type) are unknowns that have arbitrary occurrences. We bound from above the heights of terms occurring in a most general unifier in terms of the number of unknowns of not the cut type and of the height of the system. This bound yields an upper bound on the sizes of proofs in the Gentzen sequent calculus LK. Namely, we show that one can transform a proof D in LK by substituting some free terms in places of variables in such a way that the heights of terms occurring in the proof may be bounded from above by ar [D] h 1 \Delta q \Gamma [D] \Delta h 0 , where ar [D] is the maximal arity of function symbols occurring in D, h 1 is the...