Harry Lewis

• Harvard University

After two decades of progress in hardness of approximation we finally completely understand the extent to which many optimization problems can be approximated in polynomial time. Unfortunately, however, and despite significant efforts, many important ...

We survey recent results on parallel repetition theorems for computationally-sound interactive proofs (a.k.a. interactive arguments).

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Consider a family of boolean circuitsC1,C2,…,Cn,…, constructed by some uniform, effective procedure operating on inputn. Such a procedure provides a concise representation of a family of parallel algorithms for computing boolean values. A formula of first-order logic may also be viewed as a concise representation of a family of parallel algorithms...

By the word problem for some class of algebraic structures we mean the problem of determining, given a finite set E of equations between words (i.e. terms) and an additional equation x = y , whether x = y must hold in all structures satisfying each member of E . In 1947 Post [P] showed the word problem for semigroups to be undecidable. This result...

We analyze the computational complexity of determining whether F is satisfiable when F is a formula of the classical predicate calculus obeying certain syntactic restrictions. For example, for the monadic predicate calculus and the Gödel or 3 … ∃∀∀∃ … 3 prefix class we obtain lower and upper nondeterministic time bounds of the form cn/log n. The lo...

A template dependency is a formalized integrity constraint on a relational database, stating that whenever tuples exist in the database that agree on certain attributes, an additional tuple must also be present that agrees with the others in a specified way. It is shown that the inference problem for template dependencies is undecidable, that is, t...

Among the earliest and best-known theorems on the decision problem is Skolem's result [7] that the class of all closed formulas with prefixes of the form ∀···∀∃···∃ is a reduction class for satisfiability for the whole of quantification theory. This result can be refined in various ways. If the Skolem prefix alone is considered, the best result [8]...

Let Q be the class of closed quantificational formulas ∀ x ∃ u ∀ yM without identity such that M is a quantifier-free matrix containing only monadic and dyadic predicate letters and containing no atomic subformula of the form Pyx or Puy for any predicate letter P . In [DKW] Dreben, Kahr, and Wang conjectured that Q is a solvable class for satisfiab...

In this paper we consider decision problems for subclasses of Kr , the class of those formulas of pure quantification theory whose matrices are conjunctions of binary disjunctions of signed atomic formulas. If each of Q 1 , …, Q n is an ∀ or an ∃, then let Q 1 … Q n be the class of those closed prenex formulas with prefixes of the form ( Q 1 x 1 )…...

In this paper we consider classes of quantificational formulas specified by restrictions on the number of atomic subformulas appearing in a formula. Little seems to be known about the decision problem for such classes, except that the class whose members contain at most two distinct atomic subformulas is decidable [2]. (We use “decidable” and “unde...