Automating Algebraic Methods in Isabelle
... reasoning or decision procedures for data types, and integrating domain specific solvers are indispensable for these tasks. Yet all these mechanisms are available through the recent integration of ATP systems and Satisfiability Modulo Theories (SMT) solvers into Isabelle/HOL [28,4]. Our paper shows ...
... reasoning or decision procedures for data types, and integrating domain specific solvers are indispensable for these tasks. Yet all these mechanisms are available through the recent integration of ATP systems and Satisfiability Modulo Theories (SMT) solvers into Isabelle/HOL [28,4]. Our paper shows ...
Non-commutative arithmetic circuits with division
... Arithmetic circuit complexity studies the computation of polynomials and rational functions using the basic operations addition, multiplication, and division. It is chiefly interested in commutative polynomials or rational functions, defined over a set of multiplicatively commuting variables (see th ...
... Arithmetic circuit complexity studies the computation of polynomials and rational functions using the basic operations addition, multiplication, and division. It is chiefly interested in commutative polynomials or rational functions, defined over a set of multiplicatively commuting variables (see th ...
Separation of Multilinear Circuit and Formula Size
... graph with nodes of in-degree 0 or 2. We refer to the in-neighbors of a node as its “children.” Every leaf of the graph (i. e., a node of in-degree 0) is labelled with either an input variable or a field element. Every other node of the graph is labelled with either + or × (in the first case the nod ...
... graph with nodes of in-degree 0 or 2. We refer to the in-neighbors of a node as its “children.” Every leaf of the graph (i. e., a node of in-degree 0) is labelled with either an input variable or a field element. Every other node of the graph is labelled with either + or × (in the first case the nod ...
Saeed SALEHI DECIDABLE FORMULAS OF INTUITIONISTIC
... It might be expected that provably total functions of iΣn are the provably recursive functions of IΣn , but for n = 2 it is false! Burr [1] has shown that provably total functions of iΣ2 are primitive recursive, although it is well-known that the Ackermann’s function is provably recursive in IΣ2 . ...
... It might be expected that provably total functions of iΣn are the provably recursive functions of IΣn , but for n = 2 it is false! Burr [1] has shown that provably total functions of iΣ2 are primitive recursive, although it is well-known that the Ackermann’s function is provably recursive in IΣ2 . ...
Propositional Deduction via Sequent Calculus
... syntactic soundness, called consistency, which says that there is no derivation of the false formula ⊥. The approach we take here follows Gerhard Gentzen, a German mathematician of the 1930s and early 1940s.1 A Note on Greek Letters. We will use Greek letters2 such as φ, ψ, and χ (pronounced “phi,” ...
... syntactic soundness, called consistency, which says that there is no derivation of the false formula ⊥. The approach we take here follows Gerhard Gentzen, a German mathematician of the 1930s and early 1940s.1 A Note on Greek Letters. We will use Greek letters2 such as φ, ψ, and χ (pronounced “phi,” ...
Automatic differentiation
In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation or computational differentiation, is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.Automatic differentiation is not: Symbolic differentiation, nor Numerical differentiation (the method of finite differences).These classical methods run into problems: symbolic differentiation leads to inefficient code (unless carefully done) and faces the difficulty of converting a computer program into a single expression, while numerical differentiation can introduce round-off errors in the discretization process and cancellation. Both classical methods have problems with calculating higher derivatives, where the complexity and errors increase. Finally, both classical methods are slow at computing the partial derivatives of a function with respect to many inputs, as is needed for gradient-based optimization algorithms. Automatic differentiation solves all of these problems.