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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 ..."
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The kprovability problem is, given a first order formula &phi; and an integer k, to determine if &phi; 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...
Type inference and semiunification
 In Proceedings of the ACM Conference on LISP and Functional Programming (LFP ) (Snowbird
, 1988
"... In the last ten years declarationfree programming languages with a polymorphic typing discipline (ML, B) have been developed to approximate the flexibility and conciseness of dynamically typed languages (LISP, SETL) while retaining the safety and execution efficiency of conventional statically type ..."
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Cited by 32 (7 self)
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In the last ten years declarationfree programming languages with a polymorphic typing discipline (ML, B) have been developed to approximate the flexibility and conciseness of dynamically typed languages (LISP, SETL) while retaining the safety and execution efficiency of conventional statically typed languages (Algol68, Pascal). These polymorphic languages can be type checked at compile time, yet allow functions whose arguments range over a variety of types. We investigate several polymorphic type systems, the most powerful of which, termed MilnerMycroft Calculus, extends the socalled letpolymorphism found in, e.g., ML with a polymorphic typing rule for recursive definitions. We show that semiunification, the problem of solving inequalities over firstorder terms, characterizes type checking in the MilnerMycroft Calculus to polynomial time, even in the restricted case where nested definitions are disallowed. This permits us to extend some infeasibility results for related combinatorial problems to type inference and to correct several claims and statements in the literature. We prove the existence of unique most general solutions of term inequalities, called most general semiunifiers, and present an algorithm for computing them that terminates for all known inputs due to a novel “extended occurs check”. We conjecture this algorithm to be
On Herbrand Skeletons
, 1995
"... . Herbrand's theorem plays an important role both in proof theory and in computer science. Given a Herbrand skeleton, which is basically a number specifying the count of disjunctions of the matrix, we would like to get a computable bound on the size of terms which make the disjunction into ..."
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. Herbrand's theorem plays an important role both in proof theory and in computer science. Given a Herbrand skeleton, which is basically a number specifying the count of disjunctions of the matrix, we would like to get a computable bound on the size of terms which make the disjunction into a quasitautology. This is an important problem in logic, specifically in the complexity of proofs. In computer science, specifically in automated theorem proving, one hopes for an algorithm which avoids the guesses of existential substitution axioms involved in proving a theorem. Herbrand's theorem forms the very basis of automated theorem proving where for a given number n we would like to have an algorithm which finds the terms in the n disjunctions of matrices solely from the shape of the matrix. The main result of this paper is that both problems have negative solutions. 1 Introduction By the theorem of Herbrand we have for a quantifierfree OE: j= 9 x OE( x) iff j= OE( a 1 ) OE( ...
BetaReduction As Unification
, 1996
"... this report, we use a lean version of the usual system of intersection types, whichwe call . Hence, UP is also an appropriate unification problem to characterize typability of terms in . Quite apart from the new light it sheds on fireduction, such an analysis turns out to have several othe ..."
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this report, we use a lean version of the usual system of intersection types, whichwe call . Hence, UP is also an appropriate unification problem to characterize typability of terms in . Quite apart from the new light it sheds on fireduction, such an analysis turns out to have several other benefits
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 ar ..."
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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.
Generalizing Theorems in Real Closed Fields
, 1995
"... Jan Krajicek posed the following problem: Is there is a generalization result in the theory of real closed fields of the form: If A(1 + ... + 1) (n occurrences of 1) is provable in length k for all n, then (x)A(x) is provable? It is argued that the answer to this question depends on the particular f ..."
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Jan Krajicek posed the following problem: Is there is a generalization result in the theory of real closed fields of the form: If A(1 + ... + 1) (n occurrences of 1) is provable in length k for all n, then (x)A(x) is provable? It is argued that the answer to this question depends on the particular formulation of the "theory of real closed fields." Four distinct formulations are investigated with respect to their generalization behavior. It is shown that there is a positive answer to Kraj'icek's question for (1) the axiom system RCF of ArtinSchreier with Gentzen's LK as underlying logical calculus, (2) RCF with the variant LKB of LK allowing introduction of several quantifiers of the same type in one step, (3) LKB and the firstorder schemata corresponding to Dedekind cuts and the supremum principle. A negative answer is given for (4) any system containing the schema of extensionality.
Decidability and Undecidability Results for Propositional Schemata
"... We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions (e.g., pi) and iterated connectives ∨ or ∧ ranging over intervals parameterized by arithmetic variables (e.g., ∧n i=1 pi, where n is a parameter). The satisfiability problem is shown ..."
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We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions (e.g., pi) and iterated connectives ∨ or ∧ ranging over intervals parameterized by arithmetic variables (e.g., ∧n i=1 pi, where n is a parameter). The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called boundlinear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus (called stab) allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that boundlinear schemata represent a good compromise between expressivity and decidability. 1.
Propositional proof complexity  an introduction
 IN ULRICH BERGER AND HELMUT SCHWICHTENBERG, EDITORS, COMPUTATIONAL PROOF THEORY
, 1997
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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.