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The Undecidability of kProvability
 Annals of Pure and Applied Logic
, 1989
"... The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X... ..."
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The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X...
On Gödel's Theorems on Lengths of Proofs I: Number of Lines and Speedup for Arithmetics
"... This paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact, unbounded) proof speedup of (i + 1)st order arithmetic over ith order arithmetic, where arithme ..."
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Cited by 7 (0 self)
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This paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact, unbounded) proof speedup of (i + 1)st order arithmetic over ith order arithmetic, where arithmetic is formalized in Hilbertstyle calculi with + and as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higherorder logic; this allows all tautologies as axioms and allows all generalizations of axioms as axioms.
The Craig Interpolation Theorem for Schematic Systems
, 1996
"... The notion of Schematic System has been introduced by Parikh in the early seventies. It is a metamathematical notion describing the concept of deduction system and the operation of substitution of terms and formulas in it. We show a generalization of the Craig Interpolation Theorem for a natural ..."
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The notion of Schematic System has been introduced by Parikh in the early seventies. It is a metamathematical notion describing the concept of deduction system and the operation of substitution of terms and formulas in it. We show a generalization of the Craig Interpolation Theorem for a natural class of schematic systems while we determine sufficient conditions for a schematic system to enjoy Interpolation.
Interpolants, Cut Elimination and Flow Graphs . . .
, 1997
"... We analyse the structure of propositional proofs in the sequent calculus focusing on the wellknown procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to understand why a tautology might be ‘hard to prove’. Given a proof we associate to it a logical graph tracing ..."
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Cited by 4 (3 self)
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We analyse the structure of propositional proofs in the sequent calculus focusing on the wellknown procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to understand why a tautology might be ‘hard to prove’. Given a proof we associate to it a logical graph tracing the flow of formulas in it (Buss, 1991). We show some general facts about logical graphs such as acyclic @ of cutfree proofs and acyclic @ of contractionfree proofs (possibly containing cuts), and we give a proof of a strengthened version of the Craig Interpolation Theorem based on flows of formulas. We show that tautologies having minimal interpolants of nonlinear size (i.e. number of symbols) must have proofs with certain precise structural properties. We then show that given a proof ZI and a cutfree form Il ’ associated to it (obtained by a particular cut elimination procedure), certain subgraphs of II ’ which are logical graphs (i.e. graphs of proofs) correspond to subgraphs of Zl which are logical graphs for the same sequent. This locality property of cut elimination leads to new results on the complexity of interpolants, which cannot follow from the known constructions proving the Craig Interpolation Theorem.
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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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
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.
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...
A Sequent Calculus with Implicit Term Representation
, 2010
"... We investigate a modification of the sequent calculus which separates a firstorder proof into its abstract deductive structure and a unifier which renders this structure a valid proof. We define a cutelimination procedure for this calculus and show that it produces the same cutfree proofs as the s ..."
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We investigate a modification of the sequent calculus which separates a firstorder proof into its abstract deductive structure and a unifier which renders this structure a valid proof. We define a cutelimination procedure for this calculus and show that it produces the same cutfree proofs as the standard calculus, but, due to the implicit representation of terms, it provides exponentially shorter normal forms. This modified calculus is applied as a tool for theoretical analyses of the standard calculus and as a mechanism for a more efficient implementation of cutelimination.