Andreas Blass

A game semantics for linear logic

Abstract We present a game (or dialogue) semantics in the style of Lorenzen (1959) for Girards linear logic (1987). Lorenzen suggested that the (constructive) meaning of a proposition ϕ should be specified by telling how to conduct a debate between a proponent P who asserts ϕ and an opponent O who denies ϕ. Thus propositions are interpreted as games, connectives (almost) as operations on games, and validity as existence of a winning strategy for P. (The qualifier ‘almost’ will be discussed later when more details have been presented.) We propose that the connectives of linear logic can be naturally interpreted as the operations on games introduced for entirely different purposes by Blass (1972). We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of game-semantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective ⊗ is weaker than the interpretation implicit in Girards proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Godels Dialectica interpretation (1958), which was connected to linear logic by de Paiva (1989), fits with game semantics.

Combinatorial Cardinal Characteristics of the Continuum

The combinatorial study of subsets of the set N of natural numbers and of functions from N to N leads to numerous cardinal numbers, uncountable but no larger than the continuum. For example, how many infinite subsets X of N must I take so that every subset Y of N or its complement includes one of my X’s? Or how many functions f from N to N must I take so that every function from N to N is majorized by one of my f’s? The main results about these cardinal characteristics of the continuum are of two sorts: inequalities involving two (or sometimes three) characteristics, and independence results saying that other such inequalities cannot be proved in ZFC. Other results concern, for example, the cofinalities of these cardinals or connections with other areas of mathematics. This survey concentrates on the combinatorial set-theoretic aspects of the theory.

Abstract state machines capture parallel algorithms

We give an axiomatic description of parallel, synchronous algorithms. Our main result is that every such algorithm can be simulated, step for step, by an abstract state machine with a background that provides for multisets.

On the unique satisfiability problem

UNIQUE SAT is the problem of deciding whether a given Boolean formula has exactly one satisfying truth assignment. This problem is a typical (moreover complete) representative of a natural class of problems about unique solutions. All these problems belong to the class DIFe= {L1--L2:L1,Lz~NP} studied by Papadimitriou and Yannakakis. We consider the relationship between these two classes, particularly whether UNIQUE SAT is DIFe-complete: It is if NP = co- NP. We construct an oracle relative to which UNIQUE SAT is not complete for DIF ~, and another oracle relative to which UNIQUE SAT is complete for DIF e, whereas NP v ~ co - NP.

There may be simple Pℵ1 and Pℵ2-points and the Rudin-Keisler ordering may be downward directed

We prove the consistency, relative to ZFC, of each of the following two (mutually contradictory) statements. (A) Every two non-principal ultrafilters on o have a common image via a finite-to-one function. (B) Simple &,-points and simple &+-points both exist. These results, proved by the second author, answer questions of the first author and P. Nyikos, who had obtained numerous consequences of (A) and (B), respectively. In the models we construct, the bounding number is K,, while the dominating number, the splitting number, and the cardinality of the continuum are HZ.

Choiceless polynomial time

Turing machines define polynomial time (PTime) on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures. Is there a computation model whose machines do not distinguish between isomorphic structures and compute exactly PTime properties? This question can be recast as follows: Does there exist a logic that captures polynomial time (without presuming the presence of a linear order)? Earlier, one of us conjectured the negative answer. The problem motivated a quest of stronger and stronger PTime logics. All these logics avoid arbitrary choice. Here we attempt to capture the choiceless fragment of PTime. Our computation model is a version of abstract state machines (formerly called evolving algebras). The idea is to replace arbitrary choice with parallel execution. The resulting logic is more expressive than other PTime logics in the literature. A more difficult theorem shows that the logic does not capture all PTime.

Henkin quantifiers and complete problems

We analyze computational aspects of partially ordered quantification in first-order logic. show that almost any non-linear quantifier, applied to quantifier-free first-order formtt suffices to express an N~-complete predicate. The remaining non-linear quantifiers expJ exactly co-NLe predicates.

Properties of almost all graphs and complexes

There are many results in the literature asserting that almost all or almost no graphs have some property. Our object is to develop a general logical theorem that will imply almost all of these results as corollaries. To this end, we propose the first-order theory of almost all graphs by presenting Axiom n which states that for each sequence of 2n distinct vertices in a graph (u1, …, un, v1, …, vn), there exists another vertex w adjacent to each u1 and not adjacent to any vi. A simple counting argument proves that for each n, almost all graphs satisfy Axiom n. It is then shown that any sentence that can be stated in terms of these axioms is true in almost all graphs or in almost none. This has several immediate consequences, most of which have already been proved separately including: (1) For any graph H, almost all graphs have an induced subgraph isomorphic to H. (2) Almost no graphs are planar, or chordal, or line graphs. (3) Almost all grpahs are connected wiht diameter 2. It is also pointed out that these considerations extend to digraphs and to simplicial complexes.

A zero-one law for logic with a fixed-point operator

The logic obtained by adding the least-fixed-point operator to first-order logic was proposed as a query language by Aho and Ullman (in “Proc. 6th ACM Sympos. on Principles of Programming Languages,” 1979, pp. 110–120) and has been studied, particularly in connection with finite models, by numerous authors. We extend to this logic, and to the logic containing the more powerful iterative-fixed-point operator, the zero-one law proved for first-order logic in ( Glebskii, Kogan, Liogonki, and Talanov (1969) , Kibernetika 2, 31–42; Fagin (1976) , J. Symbolic Logic 41, 50–58). For any sentence φ of the extend logic, the proportion of models of φ among all structures with universe {1,2,…, n} approaches 0 or 1 as n tends to infinity. We also show that the problem of deciding, for any φ, whether this proportion approaches 1 is complete for exponential time, if we consider only φs with a fixed finite vocabulary (or vocabularies of bounded arity) and complete for double-exponential time if φ is unrestricted. In addition, we establish some related results.

Near coherence of filters. II. Applications to operator ideals, the Stone-Čech remainder of a half-line, order ideals of sequences, and slenderness of groups

The set-theoretic principle of near coherence of filters (NCF) is lnown to be neither provable nor refutable from the usual axioms of set theory. We show that NCF is equivalent to the following statements, among others: (1) The ideal of compact operators on Hilbert space is not the sum of two smaller ideals. (2) The Stone-tech remainder of a half-line has only one composant. (This was first proved by J. Mioduszewski.) (3) The partial ordering of slenderness classes of abelian groups, minus its top element, is directed upward (and in fact has a top element). Thus, 11 these statements are also consistent and independent.