Boole's Method I. A Modern Version

A rigorous, modern version of Boole's algebra of logic is presented, based partly on the 1890s treatment of Ernst Schroder

Discriminator varieties and symbolic computation

AbstractWe look at two aspects of discriminator varieties which could be of considerable interest in symbolic computation:1.discriminator varieties are unitary (i.e., there is always a most general unifier of two unifiable terms), and2.every mathematical problem can be routinely cast in the form†p1 ≈ q1, …, pk ≈ qk implies the equation x ≈ y.Item (l) offers possibilities for implementations in computational logic, and (2) shows that Birkhoff's five rules of inference for equational logic are all one needs to prove theorems in mathematics

Unification in the union of disjoint equational theories : combining decision procedures

Most of the work on the combination of unification algorithms for the union of disjoint equational theories has been restricted to algorithms which compute finite complete sets of unifiers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of unification problems without computing unifiers. In this paper we describe a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called unification problems with constant restrictions--a slight generalization of unification problems with constants--is decidable for these theories. As a consequence of this new method, we can for example show that general A-unifiability, i.e., solvability of A-unification problems with free function symbols, is decidable. Here A stands for the equational theory of one associative function symbol. Our method can also be used to combine algorithms which compute finite complete sets of unifiers. Manfred Schmidt-Schauß\u27 combination result, the until now most general result in this direction, can be obtained as a consequence of this fact. We also get the new result that unification in the union of disjoint equational theories is finitary, if general unification--i.e., unification of terms with additional free function symbols--is finitary in the single theories

Extended logic-plus-functional programming

Extensions of logic and functional programming are integrated in RELFUN. Its valued clauses comprise Horn clauses (true\u27-valued) and clauses with a distinguished foot\u27 premise (returning arbitrary values). Both the logic and functional components permit LISP-like varying-arity and higher-order operators. The DATAFUN sublanguage of the functional component is shown to be preferable to relational encodings of functions in DATALOG. RELFUN permits non-ground, non-deterministic functions, hence certain functions can be inverted using an is\u27-primitive generalizing that of PROLOG. For function nestings a strict call-by-value strategy is employed. The reduction of these extensions to a relational sublanguage is discussed and their WAM compilation is sketched. Three examples (serialise\u27, wang\u27, and eval\u27) demonstrate the relational/functional style in use. The list expressions of RELFUN\u27s LISP implementation are presented in an extended PROLOG-like syntax

Unification in monoidal theories is solving linear equations over semirings

Although for numerous equational theories unification algorithms have been developed there is still a lack of general methods. In this paper we apply algebraic techniques to the study of a whole class of theories, which we call monoidal. Our approach leads to general results on the structure of unification algorithms and the unification type of such theories. An equational theory is monoidal if it contains a binary operation which is associative and commutative, an identity for the binary operation, and an arbitrary number of unary symbols which are homomorphisms for the binary operation and the identity. Monoidal theories axiomatize varieties of abelian monoids. Examples are the theories of abelian monoids (AC), idempotent abelian monoids (ACI), and abelian groups. To every monoidal theory we associate a semiring. Intuitively, semirings are rings without subtraction. We show that every unification problem in a monoidal theory can be translated into a system of linear equations over the corresponding semiring. More specifically, problems without free constants are translated into homogeneous equations. For problems with free constants inhomogeneous equations have to be solved in addition. Exploiting the correspondence between unification and linear algebra we give algebraic characterizations of the unification type of a theory. In particular, we show that with respect to unification without constants monoidal theories are either unitary or nullary. Applying Hilbert\u27s Basis Theorem we prove that theories of groups with commuting homomorphisms are unitary with respect to unification with and without constants

Deterministic stream-sampling for probabilistic programming: semantics and verification

Probabilistic programming languages rely fundamentally on some notion of sampling, and this is doubly true for probabilistic programming languages which perform Bayesian inference using Monte Carlo techniques. Verifying samplers - proving that they generate samples from the correct distribution - is crucial to the use of probabilistic programming languages for statistical modelling and inference. However, the typical denotational semantics of probabilistic programs is incompatible with deterministic notions of sampling. This is problematic, considering that most statistical inference is performed using pseudorandom number generators.We present a higher-order probabilistic programming language centred on the notion of samplers and sampler operations. We give this language an operational and denotational semantics in terms of continuous maps between topological spaces. Our language also supports discontinuous operations, such as comparisons between reals, by using the type system to track discontinuities. This feature might be of independent interest, for example in the context of differentiable programming.Using this language, we develop tools for the formal verification of sampler correctness. We present an equational calculus to reason about equivalence of samplers, and a sound calculus to prove semantic correctness of samplers, i.e. that a sampler correctly targets a given measure by construction