gapt.examples

Creates the n-th tautology of a sequence that has only exponential-size cut-free proofs

This sequence is taken from: S. Buss. "Weak Formal Systems and Connections to Computational Complexity". Lecture Notes for a Topics Course, UC Berkeley, 1988, available from: http://www.math.ucsd.edu/~sbuss/ResearchWeb/index.html

Sequence of valid first-order formulas about equivalent counting methods.

Consider the formula ∀z ∃^=1^i ∀x ∃y a_i(x,y,z), where ∃^=1^i is a quantifier that says that there exists exactly one i (in 0..n) such that ∀x ∃y a_i(x,y,z) is true.

This function returns the equivalence between two implementations of the formula: first, using a naive quadratic implementation; and second, using an O(n*log(n)) implementation with threshold formulas.

Contains some commonly used formulas.

Monoid cancellation benchmark from Gregory Malecha and Jesper Bengtson: Extensible and Efficient Automation Through Reflective Tactics, ESOP 2016.

Creates the n-th formula of a sequence where distributivity-based algorithm produces only exponential CNFs.

Given n >= 2 creates an unsatisfiable first-order clause set based on a statement about the permutations in S_n.

Constructs a formula representing the pigeon hole principle. More precisely: PigeonHolePrinciple( p, h ) states that if p pigeons are put into h holes then there is a hole which contains two pigeons. PigeonHolePrinciple( p, h ) is a tautology iff p > h.

Since we want to avoid empty disjunctions, we assume > 1 pigeons.

Provides a simple intuitionistic proof of ¬p ∨ p ⊢ ¬¬p → p. Applying the CERES method will create a non-intuitionistic proof but reductive cut-elimination will always create an intuitionistic one. Therefore this is an example that CERES produces cut-free proofs which reductive cut-elimination cannot.

Version 2 of the higher-order n-Tape proof.

Version 3 of the higher-order n-Tape proof.

Version 3 of the higher-order n-Tape proof.

Version 5 of the higher-order n-Tape proof, where if-then-else is directly axiomatized i.e. it has 2 additional axioms P -> if code(P) then t else f = t and -P -> if code(P) then t else f = f which were theorems before. In contrast to nTape4 it cuts on instances of the theorem C for specific upper bounds n. Since the instantiated proofs were generated manually, only nTape5(2) to nTape5(4) work.

Version 5 of the higher-order n-Tape proof. In contrast to nTape4 it cuts on instances of the theorem C for specific upper bounds n. Since the instantiated proofs were generated manually, only nTape5(2) to nTape5(4) work.

Version 5 of the higher-order n-Tape proof, where if-then-else is still proved in arithmetic. In contrast to nTape4 it cuts on instances of the theorem C for specific upper bounds n. Since the instantiated proofs were generated manually, only nTape5Arith(2) works.

Version 5 of the higher-order n-Tape proof, where if-then-else is still proved in arithmetic. In contrast to nTape4 it cuts on instances of the theorem C for specific upper bounds n. Since the instantiated proofs were generated manually, only nTape5Arith(2) works.

The object nTape6 generates hard problems for higher order theorem provers containing an axiomatization of if-then-else. Formulas: f1,f2 ... if-then-else axiomatizations f3,f4 ... properties of the successor function (0 is no successor and a number is always different from its successor) conclusion0 ... there exists a function h s.t. h(0) = 1, h(1) = 0 conclusion1 ... there exists a function h s.t. h(0) = 1, h(1) = 0, h(2) = 0 conclusion2 ... there exists a function h s.t. h(0) = 1, h(1) = 0, h(2) = 1 w1 ... witness for sc w2 ... witness for sc2

The problems are (in sequent notation):

P0: f1, f2 :- conclusion0 P1: f1, f2, f3, f4 :- conclusion1 P2: f1, f2, f3, f4 :- conclusion2

The generated filenames are "ntape6-i-without-witness.tptp" for i = 0 to 2.

To show that there are actual witnesses for the function h, we provide a witness, where the witness w1 can be used for both W0 and W1:

W0: { w1 :- } x P0 W1: { w1 :- } x P1 W2: { w2 :- } x P2

The generated filenames are "ntape6-i-with-witness.tptp" for i = 0 to 2.

Formalisation of the tape-proof as described in C. Urban: Classical Logic and Computation, PhD Thesis, Cambridge University, 2000.

This is an example used in the talk[1] at TbiLLC 2013. It generates a (cut-free) LK proof where the extracted expansion tree has nested quantifiers.

[1] http://www.illc.uva.nl/Tbilisi/Tbilisi2013/uploaded_files/inlineitem/riener.pdf