This paper is concerned with the complexity analysis of constructor term rewrite systems and its ramification in implicit computational complexity. We introduce a path order with multiset status, the polynomial path order POP ∗ , that is applicable in two related, but distinct contexts. On the one hand POP ∗ induces polynomial innermost runtime complexity and hence may serve as a syntactic, and fully automatable, method to analyse the innermost runtime complexity of term rewrite systems. On the other hand POP ∗ provides an order-theoretic characterisation of the polytime computable functions: the polytime computable functions are exactly the functions computable by an orthogonal constructor TRS compatible with POP ∗ .
There are several accounts of predicative analysis of recursion in the (ICC) literature. We mention Marion's ( for short) @cite_26 . The path order provides an order-theoretic characterisation of the class @math and can be also consider as a miniaturisation of of sorts: it is a restriction of and yields an order-theoretic characterisation of a complexity class. On the other hand cannot be used to characterise the (innermost) runtime complexity of TRSs. This follows from Example below. In particular, although @math is compatible with @math , from this we can only conclude that @math is computable on a nondeterministic Turing machine in polynomial time. However, this follows by design as @math is complete for @math . For a precedence @math that fulfils @math and @math we obtain that @math is compatible with @math . However it is straightforward to verify that the family of terms @math admits (innermost) derivations whose length grows exponentially in @math . Still the underlying function can be proven polynomial, essentially relying on memoisation techniques @cite_26 .
@cite_26
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Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. A natural term rewriting framework for the Bellantoni Cook schemata of predicative recursion, which yields a canonical definition of the polynomial time computable functions, is introduced. In terms of an exponential function both, an upper bound and a lower bound are proved for the resulting derivation lengths of the functions in question. It is proved that any natural reduction strategy yields an algorithm which runs in exponential time. We give an example in which this estimate is tight. It is proved that the resulting derivation lengths become polynomially bounded in the lengths of the inputs if the rewrite rules are only applied to terms in which the safe arguments – no restrictions are assumed for the normal arguments – consist of values, i.e. numerals, and not of names, i.e. non numeral terms. It is proved that in the latter situation any inside first reduction strategy and any head reduction strategy yield algorithms, for the function under consideration, for which the running time is bounded by an appropriate polynomial in the lengths of the input. A feasible rewrite system for predicative recursion with predicative parameter substitution is defined. It is proved that the derivation lengths of this rewrite system are polynomially bounded in the lengths of the inputs. As a corollary we reobtain Bellantoni’s result stating that predicative recursion is closed under predicative parameter recursion. Let @math be a @math -ary predicate over a finite alphabet. Consider a random CSP @math instance @math over @math variables with @math constraints. When @math the instance @math will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When @math is the @math -ary OR predicate, this is the well studied problem of refuting random @math -SAT formulas, and an efficient algorithm is known only when @math . Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a @math -ary predicate, having @math constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if @math fails to support a @math -wise uniform distribution, then there is an efficient algorithm that refutes random CSP @math instances @math whp when @math . Indeed, our algorithm will "somewhat strongly" refute @math , certifying @math , if @math then we get the strongest possible refutation, certifying @math . This last result is new even in the context of random @math -SAT. Regarding the optimality of our @math requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP @math may be impossible when @math . Thus there is an indication our algorithm's dependence on @math is optimal for every @math , at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the @math -round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz. We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math .