Albert R. Meyer

Word problems requiring exponential time(Preliminary Report)

The equivalence problem for Kleenes regular expressions has several effective solutions, all of which are computationally inefficient. In [1], we showed that this inefficiency is an inherent property of the problem by showing that the problem of membership in any arbitrary context-sensitive language was easily reducible to the equivalence problem for regular expressions. We also showed that with a squaring abbreviation ( writing (E)2 for E×E) the equivalence problem for expressions required computing space exponential in the size of the expressions. In this paper we consider a number of similar decidable word problems from automata theory and logic whose inherent computational complexity can be precisely characterized in terms of time or space requirements on deterministic or nondeterministic Turing machines. The definitions of the word problems and a table summarizing their complexity appears in the next section. More detailed comments and an outline of some of the proofs follows in the remaining sections. Complete proofs will appear in the forthcoming papers [9, 10, 13]. In the final section we describe some open problems.

The complexity of the word problems for commutative semigroups and polynomial ideals

Abstract Any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied. This bound is achieved by a simple procedure for the semigroup problem.

Counter machines and counter languages

The languages recognizable by time- and space-restricted multiple-counter machines are compared to the languages recognizable by similarly restricted multipletape Turing machines. Special emphasis is placed on languages definable by machines which operate in “real time”. Time and space requirements for counter machines and Turing machines are analyzed. A number of questions which remain open for time-restricted Turing machines are settled for their counter machine counterparts.

Coping with errors in binary search procedures

Abstract We consider the problem of identifying an unknown value x ϵ {1, 2,…, n } using only comparisons of x to constants when as many as E of the comparisons may receive erroneous answers. For a continuous analogue of this problem we show that there is a unique strategy that is optimal in the worst case. This strategy for the continuous problem is then shown to yield a strategy for the original discrete problem that uses log 2 n + E · log 2 log 2 n + O ( E · log 2 E ) comparisons in the worst case. This number is shown to be optimal even if arbitrary “Yes-No” questions are allowed. We show that a modified version of this search problem with errors is equivalent to the problem of finding the minimal root of a set of increasing functions. The modified version is then also shown to be of complexity log 2 n + E · log 2 log 2 n + O ( E · log 2 E ).

The complexity of loop programs

Anyone familiar with the theory of computability will be aware that practical conclusions from the theory must be drawn with caution. If a problem can theoretically be solved by computation, this does not mean that it is practical to do so. Conversely, if a problem is formally undecidable, this does not mean that the subcases of primary interest are impervious to solution by algorithmic methods. In the next section we describe such a class of programs, called “Loop programs.” Each Loop program consists only of assignment statements and iteration (loop) statements, the latter resembling the DO statement of FORTRAN, and special cases of the FOR and THROUGH statements of ALGOL and MAD. The bound on the running time of a Loop program is determined essentially by the length of the program and the depth of nesting of its loops.

What is a model of the lambda calculus

An elementary, purely algebraic definition of model for the untyped lambda calculus is given. This definition is shown to be equivalent to the natural semantic definition based on environments. These definitions of model are consistent with, and yield a completeness theorem for, the standard axioms for lambda convertibility. A simple construction of models for lambda calculus is reviewed. The algebraic formulation clarifies the relation between combinators and lambda terms.

Real-Time Simulation of Multihead Tape Units

The main result of this paper is that, given a Turing machine with several read-write heads per tape, one can effectively construct an equivalent multitape Turing machine with a single read-write head per tape, which runs at precisely the same speed. This result implies that serial storage may be used to handle files requiring several points of immediate two-way read-write access without interruptions for rewinds, etc. It also yields simplified proofs of several results in the literature of computational complexity.

Towards fully abstract semantics for local variables

The Store Model of Halpern-Meyer-Trakhtenbrot is shown—after suitable repair—to be a fully abstract model for a limited fragment of ALGOL in which procedures do not take procedure parameters. A simple counter-example involving a parameter of program type shows that the model is not fully abstract in general. Previous proof systems for reasoning about procedures are typically sound for the HMT store model, so it follows that theorems about the counter-example are independent of such proof systems. Based on a generalization of standard cpo based models to structures called locally complete partial orders (lcpos), improved models and stronger proof rules are developed to handle such examples.

Exponential space complete problems for Petri nets and commutative semigroups (Preliminary Report)

The uniform word problem for commutative semigroups (UWCS) is the problem of determining from any given finite set of defining relations and any pair of words, whether the words describe the same element in the commutative semigroup defined by the relations. The effective decidability of this classical algebraic problem was first explicitly noted by Malcev [1958] and Emilichev [1958], though in retrospect this result can be seen to be contained in the earlier work of König [1903] and Hermann [1926] on polynomial ideals.

The semantics of second-order lambda calculus

Abstract In the second-order (polymorphic) typed lambda calculus, lambda abstraction over type variables leads to terms denoting polymorphic functions. Straightforward cardinality considerations show that a naive set-theoretic interpretation of the calculus is impossible. We give two definitions of semantic models for this language and prove them equivalent. Our syntactical “environment model” definition and a more algebraic “combinatory model” definition for the polymorphic calculus correspond to analogous model definitions for untyped lambda calculus. Soundness and completeness theorems are proved using the environment model definition. We verify that some specific interpretations of the calculus proposed in the literature indeed yield models in our sense.