A poset classifying non-commutative term orders

We study a certain poset on the free monoid on a countable alphabet. This poset is determined by the fact that its total extensions are precisely the standard term orders. We also investigate the poset classifying degree-compatible standard term orders, and the poset classifying sorted term orders. For the latter poset, we give a Galois coconnection with the Young lattice

Termination analysis based on operational semantics

In principle termination analysis is easy: find a well-founded partial order and prove that calls decrease with respect to this order. In practice this often requires an oracle (or a theorem prover) for determining the well-founded order and this oracle may not be easily implementable. Our approach circumvents some of these problems by exploiting the inductive definition of algebraic data types and using pattern matching as in functional languages. We develop a termination analysis for a higher-order functional language; the analysis incorporates and extends polymorphic type inference and axiomatizes a class of well-founded partial orders for multiple-argument functions (as in Standard ML and Miranda). Semantics is given by means of operational (natural-style) semantics and soundness is proved; this involves making extensions to the semantic universe and we relate this to the techniques of denotational semantics. For dealing with the partiality aspects of the soundness proof, it suffices to incorporate approximations to the desired fixed points; for dealing with the totality aspects of the soundness proof, we also have to incorporate functions that are forced to terminate (in a way that might violate the monotonicity of denotational semantics)

The Order Types of Termination Orderings on Monadic Terms, Strings and Multisets

We consider total well-founded orderings on monadic terms satisfying the replacement and full invariance properties. We show that any such ordering on monadic terms in one variable and two unary function symbols must have order type omega, omega(2) or omega(omega). We show that a familiar construction gives rise to continuum many such orderings of order type omega. We construct a new family of such orderings of order type omega(2), and show that there are continuum many of these. We show that there are only four such orderings of order type omega(omega), the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings on N-n which are preserved under vector addition. we show: that any such ordering must have order type omega(k) for some 1 less than or equal to k less than or equal to n. We show that if k &lt; tr there are continuum many such orderings, and if k = n there are only n!, the n! lexicographic orderings.</p

The Order Types of Termination Orderings on Monadic Terms, Strings and Multisets

We consider total well-founded orderings on monadic terms satisfying the replacement and full invariance properties. We show that any such ordering on monadic terms in one variable and two unary function symbols must have order type omega, omega(2) or omega(omega). We show that a familiar construction gives rise to continuum many such orderings of order type omega. We construct a new family of such orderings of order type omega(2), and show that there are continuum many of these. We show that there are only four such orderings of order type omega(omega), the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings on N-n which are preserved under vector addition. we show: that any such ordering must have order type omega(k) for some 1 less than or equal to k less than or equal to n. We show that if k &lt; tr there are continuum many such orderings, and if k = n there are only n!, the n! lexicographic orderings.</p

Two applications of analytic functors

AbstractWe apply the theory of analytic functors to two topics related to theoretical computer science. One is a mathematical foundation of certain syntactic well-quasi-orders and well-orders appearing in graph theory, the theory of term rewriting systems, and proof theory. The other is a new verification of the Lagrange–Good inversion formula using several ideas appearing in semantics of lambda calculi, especially the relation between categorical traces and fixpoint operators